Riemann sums (left, right, and midpoint evaluation points) - AP Calculus AB

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Question

Estimate the integral of from 0 to 3 using left-Riemann sum and 6 rectangles. Use $\Delta x=0.5$

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Answer

Because our $\Delta x$ is constant, the left Riemann sum will be

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Question

$\begin{align}&$\text{The function }$f(x)$\text{ has the following values on the interval }$x=[-2,13]:\&f(-2)=5\&f(1)=-19\&f(4)=5\&f(7)=-6\&f(10)=-18\&f(13)=0\&\&$\text{Approximate the integral of }$f(x)$\text{ on this interval using right Riemann sums.}$\end{align}$

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Answer

$\begin{align}&$\text{A Riemann sum integral approximation over an interval [a, b]}$\&$\text{with n subintervals follows the form:}$\&\int_$a^b$ f(x)dx \approx $\frac{b-a}{n}$(f(x_1)+f(x_2)+...+f(x_n))\&$\text{It is essentially a sum of n rectangles, each with a base of}$\&$\text{length :}$$\frac{b-a}{n}$$\text{ and variable heights :}$f(x_i)$\text{, which depend on the function value at }$x_i\&$\text{In our case, we have function values over an interval:}$\&f(-2)=5\&f(1)=-19\&f(4)=5\&f(7)=-6\&f(10)=-18\&f(13)=0\&\&$\text{Although the total interval has a length of }$15\&$\text{Notice the points are equidistant. This distance is our subinterval }$3\&$\text{and since we are using the right point of each interval, we disregard the first function value:}$\&\int_${-2}^{13}$f(x)\approx3[(-19)+(5)+(-6)+(-18)+(0)]\&\int_${-2}^{13}$f(x)\approx-114\end{align}$

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Question

$\begin{align}&$\text{Approximate }$\int_${-8}^{1}$f(x)\&$\text{Using a right Riemann sum, if }$f(x)$\text{ has the values:}$\&f(-8)=-13\&f(-5)=-19\&f(-2)=6\&f(1)=-9\end{align}$

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$\begin{align}&$\text{A Riemann sum integral approximation over an interval [a, b]}$\&$\text{with n subintervals follows the form:}$\&\int_$a^b$ f(x)dx \approx $\frac{b-a}{n}$(f(x_1)+f(x_2)+...+f(x_n))\&$\text{It is essentially a sum of n rectangles, each with a base of}$\&$\text{length :}$$\frac{b-a}{n}$$\text{ and variable heights :}$f(x_i)$\text{, which depend on the function value at }$x_i\&$\text{In our case, we have function values over an interval:}$\&f(-8)=-13\&f(-5)=-19\&f(-2)=6\&f(1)=-9\&$\text{Although the total interval has a length of }$9\&$\text{Notice the points are equidistant. This distance is our subinterval: }$3\&$\text{Since we are using the right point of each interval, we disregard the first function value:}$\&\int_${-8}^{1}$f(x)\approx3[(-19)+(6)+(-9)]\&\int_${-8}^{1}$f(x)\approx-66\end{align}$

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Question

$\begin{align}&$\text{Approximate }$\int_${-2}^{23}$f(x)\&$\text{Using a right Riemann sum, if }$f(x)$\text{ has the values:}$\&f(-2)=-2\&f(3)=-15\&f(8)=14\&f(13)=15\&f(18)=-9\&f(23)=-12\end{align}$

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Answer

$\begin{align}&$\text{A Riemann sum integral approximation over an interval [a, b]}$\&$\text{with n subintervals follows the form:}$\&\int_$a^b$ f(x)dx \approx $\frac{b-a}{n}$(f(x_1)+f(x_2)+...+f(x_n))\&$\text{It is essentially a sum of n rectangles, each with a base of}$\&$\text{length :}$$\frac{b-a}{n}$$\text{ and variable heights :}$f(x_i)$\text{, which depend on the function value at }$x_i\&$\text{In our case, we have function values over an interval:}$\&f(-2)=-2\&f(3)=-15\&f(8)=14\&f(13)=15\&f(18)=-9\&f(23)=-12\&$\text{Although the total interval has a length of }$25\&$\text{Notice the points are equidistant, with }$5$\text{ intervals between. Each equidistant interval has length: }$5\&$\text{Since we are using the right point of each interval, we disregard the first function value:}$\&\int_${-2}^{23}$f(x)\approx5[(-15)+(14)+(15)+(-9)+(-12)]\&\int_${-2}^{23}$f(x)\approx-35\end{align}$

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$\begin{align}&$\text{Approximate }$\int_${-7}^{13}$f(x)\&$\text{Using a right Riemann sum, if }$f(x)$\text{ has the values:}$\&f(-7)=18\&f(-2)=-17\&f(3)=-16\&f(8)=-15\&f(13)=-14\end{align}$

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Answer

$\begin{align}&$\text{A Riemann sum integral approximation over an interval [a, b]}$\&$\text{with n subintervals follows the form:}$\&\int_$a^b$ f(x)dx \approx $\frac{b-a}{n}$(f(x_1)+f(x_2)+...+f(x_n))\&$\text{It is essentially a sum of n rectangles, each with a base of}$\&$\text{length :}$$\frac{b-a}{n}$$\text{ and variable heights :}$f(x_i)$\text{, which depend on the function value at }$x_i\&$\text{In our case, we have function values over an interval:}$\&f(-7)=18\&f(-2)=-17\&f(3)=-16\&f(8)=-15\&f(13)=-14\&$\text{Although the total interval has a length of }$20\&$\text{, notice the points are equidistant, with }$4$\text{ intervals between.}$\&$\text{Each equidistant interval has length: }$5\&$\text{Since we are using the right point of each interval, we disregard the first function value:}$\&\int_${-7}^{13}$f(x)\approx5[(-17)+(-16)+(-15)+(-14)]\&\int_${-7}^{13}$f(x)\approx-310\end{align}$

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Question

$\begin{align}&f(4)=9\&f(6)=10\&f(8)=-18\&f(10)=15\&f(12)=18\&$\text{Approximate, using a right Riemann sum, }$\int_${4}^{12}$f(x)\end{align}$

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Answer

$\begin{align}&$\text{A Riemann sum integral approximation over an interval [a, b]}$\&$\text{with n subintervals follows the form:}$\&\int_$a^b$ f(x)dx \approx $\frac{b-a}{n}$(f(x_1)+f(x_2)+...+f(x_n))\&$\text{It is essentially a sum of n rectangles, each with a base of}$\&$\text{length :}$$\frac{b-a}{n}$$\text{ and variable heights :}$f(x_i)$\text{, which depend on the function value at }$x_i\&$\text{In our case, we have function values over an interval:}$\&f(4)=9\&f(6)=10\&f(8)=-18\&f(10)=15\&f(12)=18\&$\text{Although the total interval has a length of }$8\&$\text{, notice the points are equidistant, with }$4$\text{ intervals between.}$\&$\text{Each equidistant interval has length: }$2\&$\text{Since we are using the right point of each interval, we disregard the first function value:}$\&\int_${4}^{12}$f(x)\approx2[(10)+(-18)+(15)+(18)]\&\int_${4}^{12}$f(x)\approx50\end{align}$

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Question

$\begin{align}&$\text{The function }$f(x)$\text{ has the following values on the interval }$x=[-2,38]:\&f(-2)=-13\&f(6)=-4\&f(14)=-15\&f(22)=-19\&f(30)=18\&f(38)=-8\&$\text{Approximate the integral of }$f(x)$\text{ on this interval using right Riemann sums.}$\end{align}$

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Answer

$\begin{align}&$\text{A Riemann sum integral approximation over an interval [a, b]}$\&$\text{with n subintervals follows the form:}$\&\int_$a^b$ f(x)dx \approx $\frac{b-a}{n}$(f(x_1)+f(x_2)+...+f(x_n))\&$\text{It is essentially a sum of n rectangles, each with a base of}$\&$\text{length :}$$\frac{b-a}{n}$$\text{ and variable heights :}$f(x_i)$\text{, which depend on the function value at }$x_i\&$\text{In our case, we have function values over an interval:}$\&f(-2)=-13\&f(6)=-4\&f(14)=-15\&f(22)=-19\&f(30)=18\&f(38)=-8\&$\text{Although the total interval has a length of }$40\&$\text{, notice the points are equidistant, with }$5$\text{ intervals between.}$\&$\text{Each equidistant interval has length: }$8\&$\text{Since we are using the right point of each interval, we disregard the first function value:}$\&\int_${-2}^{38}$f(x)\approx8[(-4)+(-15)+(-19)+(18)+(-8)]\&\int_${-2}^{38}$f(x)\approx-224\end{align}$

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Question

$\begin{align}&$\text{Approximate }$\int_${0}^{15}$f(x)\&$\text{Using a right Riemann sum, if }$f(x)$\text{ has the values:}$\&f(0)=-19\&f(5)=14\&f(10)=2\&f(15)=15\end{align}$

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Answer

$\begin{align}&$\text{A Riemann sum integral approximation over an interval [a, b]}$\&$\text{with n subintervals follows the form:}$\&\int_$a^b$ f(x)dx \approx $\frac{b-a}{n}$(f(x_1)+f(x_2)+...+f(x_n))\&$\text{It is essentially a sum of n rectangles, each with a base of}$\&$\text{length :}$$\frac{b-a}{n}$$\text{ and variable heights :}$f(x_i)$\text{, which depend on the function value at }$x_i\&$\text{In our case, we have function values over an interval:}$\&f(0)=-19\&f(5)=14\&f(10)=2\&f(15)=15\&$\text{Although the total interval has a length of }$15\&$\text{, notice the points are equidistant, with }$3$\text{ intervals between.}$\&$\text{Each equidistant interval has length: }$5\&$\text{Since we are using the right point of each interval, we disregard the first function value:}$\&\int_${0}^{15}$f(x)\approx5[(14)+(2)+(15)]\&\int_${0}^{15}$f(x)\approx155\end{align}$

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Question

$\begin{align}&$\text{Approximate }$\int_${-8}^{0}$f(x)\&$\text{Using a right Riemann sum, if }$f(x)$\text{ has the values:}$\&f(-8)=7\&f(-6)=-7\&f(-4)=16\&f(-2)=-16\&f(0)=20\end{align}$

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$\begin{align}&$\text{A Riemann sum integral approximation over an interval [a, b]}$\&$\text{with n subintervals follows the form:}$\&\int_$a^b$ f(x)dx \approx $\frac{b-a}{n}$(f(x_1)+f(x_2)+...+f(x_n))\&$\text{It is essentially a sum of n rectangles, each with a base of}$\&$\text{length :}$$\frac{b-a}{n}$$\text{ and variable heights :}$f(x_i)$\text{, which depend on the function value at }$x_i\&$\text{In our case, we have function values over an interval:}$\&f(-8)=7\&f(-6)=-7\&f(-4)=16\&f(-2)=-16\&f(0)=20\&$\text{Although the total interval has a length of }$8\&$\text{, notice the points are equidistant, with }$4$\text{ intervals between.}$\&$\text{Each equidistant interval has length: }$2\&$\text{Since we are using the right point of each interval, we disregard the first function value:}$\&\int_${-8}^{0}$f(x)\approx2[(-7)+(16)+(-16)+(20)]\&\int_${-8}^{0}$f(x)\approx26\end{align}$

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Question

$\begin{align}&$\text{Approximate }$\int_${-6}^{39}$f(x)\&$\text{Using a right Riemann sum, if }$f(x)$\text{ has the values:}$\&f(-6)=-4\&f(3)=-1\&f(12)=11\&f(21)=13\&f(30)=-16\&f(39)=-13\end{align}$

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Answer

$\begin{align}&$\text{A Riemann sum integral approximation over an interval [a, b]}$\&$\text{with n subintervals follows the form:}$\&\int_$a^b$ f(x)dx \approx $\frac{b-a}{n}$(f(x_1)+f(x_2)+...+f(x_n))\&$\text{It is essentially a sum of n rectangles, each with a base of}$\&$\text{length :}$$\frac{b-a}{n}$$\text{ and variable heights :}$f(x_i)$\text{, which depend on the function value at }$x_i\&$\text{In our case, we have function values over an interval:}$\&f(-6)=-4\&f(3)=-1\&f(12)=11\&f(21)=13\&f(30)=-16\&f(39)=-13\&$\text{Although the total interval has a length of }$45\&$\text{, notice the points are equidistant, with }$5$\text{ intervals between.}$\&$\text{Each equidistant interval has length: }$9\&$\text{Since we are using the right point of each interval, we disregard the first function value:}$\&\int_${-6}^{39}$f(x)\approx9[(-1)+(11)+(13)+(-16)+(-13)]\&\int_${-6}^{39}$f(x)\approx-54\end{align}$

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Question

$\begin{align}&$\text{Approximate }$\int_${-5}^{13}$f(x)\&$\text{Using a right Riemann sum, if }$f(x)$\text{ has the values:}$\&f(-5)=-13\&f(1)=-12\&f(7)=17\&f(13)=7\end{align}$

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Answer

$\begin{align}&$\text{A Riemann sum integral approximation over an interval [a, b]}$\&$\text{with n subintervals follows the form:}$\&\int_$a^b$ f(x)dx \approx $\frac{b-a}{n}$(f(x_1)+f(x_2)+...+f(x_n))\&$\text{It is essentially a sum of n rectangles, each with a base of}$\&$\text{length :}$$\frac{b-a}{n}$$\text{ and variable heights :}$f(x_i)$\text{, which depend on the function value at }$x_i\&$\text{In our case, we have function values over an interval:}$\&f(-5)=-13\&f(1)=-12\&f(7)=17\&f(13)=7\&$\text{Although the total interval has a length of }$18\&$\text{, notice the points are equidistant, with }$3$\text{ intervals between.}$\&$\text{Each equidistant interval has length: }$6\&$\text{Since we are using the right point of each interval, we disregard the first function value:}$\&\int_${-5}^{13}$f(x)\approx6[(-12)+(17)+(7)]\&\int_${-5}^{13}$f(x)\approx72\end{align}$

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Question

$\begin{align}&$\text{The function }$f(x)$\text{ has the following values on the interval }$x=[1,11]:\&f(1)=10\&f(3)=3\&f(5)=-13\&f(7)=4\&f(9)=-8\&f(11)=-15\&$\text{Approximate the integral of }$f(x)$\text{ on this interval using right Riemann sums.}$\end{align}$

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$\begin{align}&$\text{A Riemann sum integral approximation over an interval [a, b]}$\&$\text{with n subintervals follows the form:}$\&\int_$a^b$ f(x)dx \approx $\frac{b-a}{n}$(f(x_1)+f(x_2)+...+f(x_n))\&$\text{It is essentially a sum of n rectangles, each with a base of}$\&$\text{length :}$$\frac{b-a}{n}$$\text{ and variable heights :}$f(x_i)$\text{, which depend on the function value at }$x_i\&$\text{In our case, we have function values over an interval:}$\&f(1)=10\&f(3)=3\&f(5)=-13\&f(7)=4\&f(9)=-8\&f(11)=-15\&$\text{Although the total interval has a length of }$10\&$\text{, notice the points are equidistant, with }$5$\text{ intervals between.}$\&$\text{Each equidistant interval has length: }$2\&$\text{Since we are using the right point of each interval, we disregard the first function value:}$\&\int_${1}^{11}$f(x)\approx2[(3)+(-13)+(4)+(-8)+(-15)]\&\int_${1}^{11}$f(x)\approx-58\end{align}$

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Question

$\begin{align}&$\text{The function }$f(x)$\text{ has the following values on the interval }$x=[-7,3]:\&f(-7)=-18\&f(-5)=-2\&f(-3)=-20\&f(-1)=16\&f(1)=-12\&f(3)=-17\&$\text{Approximate the integral of }$f(x)$\text{ on this interval using right Riemann sums.}$\end{align}$

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$\begin{align}&$\text{A Riemann sum integral approximation over an interval [a, b]}$\&$\text{with n subintervals follows the form:}$\&\int_$a^b$ f(x)dx \approx $\frac{b-a}{n}$(f(x_1)+f(x_2)+...+f(x_n))\&$\text{It is essentially a sum of n rectangles, each with a base of}$\&$\text{length :}$$\frac{b-a}{n}$$\text{ and variable heights :}$f(x_i)$\text{, which depend on the function value at }$x_i\&$\text{In our case, we have function values over an interval:}$\&f(-7)=-18\&f(-5)=-2\&f(-3)=-20\&f(-1)=16\&f(1)=-12\&f(3)=-17\&$\text{Although the total interval has a length of }$10\&$\text{, notice the points are equidistant, with }$5$\text{ intervals between.}$\&$\text{Each equidistant interval has length: }$2\&$\text{Since we are using the right point of each interval, we disregard the first function value:}$\&\int_${-7}^{3}$f(x)\approx2[(-2)+(-20)+(16)+(-12)+(-17)]\&\int_${-7}^{3}$f(x)\approx-70\end{align}$

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Question

$\begin{align}&$\text{Approximate }$\int_${5}^{13}$f(x)\&$\text{Using a left Riemann sum, if }$f(x)$\text{ has the values:}$\&f(5)=-7\&f(7)=-8\&f(9)=-18\&f(11)=-8\&f(13)=-19\end{align}$

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Answer

$\begin{align}&$\text{A Riemann sum integral approximation over an interval [a, b]}$\&$\text{with n subintervals follows the form:}$\&\int_$a^b$ f(x)dx \approx $\frac{b-a}{n}$(f(x_1)+f(x_2)+...+f(x_n))\&$\text{It is essentially a sum of n rectangles, each with a base of}$\&$\text{length :}$$\frac{b-a}{n}$$\text{ and variable heights :}$f(x_i)$\text{, which depend on the function value at }$x_i\&$\text{In our case, we have function values over an interval:}$\&f(5)=-7\&f(7)=-8\&f(9)=-18\&f(11)=-8\&f(13)=-19\&$\text{Although the total interval has a length of }$8\&$\text{, notice the points are equidistant, with }$4$\text{ intervals between.}$\&$\text{Each equidistant interval has length: }$2\&$\text{Since we are using the left point of each interval, we disregard the larst function value:}$\&\int_${5}^{13}$f(x)\approx2[(-7)+(-8)+(-18)+(-8)]\&\int_${5}^{13}$f(x)\approx-82\end{align}$

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$\begin{align}&$\text{Approximate }$\int_${-9}^{26}$f(x)\&$\text{Using a left Riemann sum, if }$f(x)$\text{ has the values:}$\&f(-9)=-17\&f(-2)=11\&f(5)=17\&f(12)=1\&f(19)=-16\&f(26)=13\end{align}$

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Answer

$\begin{align}&$\text{A Riemann sum integral approximation over an interval [a, b]}$\&$\text{with n subintervals follows the form:}$\&\int_$a^b$ f(x)dx \approx $\frac{b-a}{n}$(f(x_1)+f(x_2)+...+f(x_n))\&$\text{It is essentially a sum of n rectangles, each with a base of}$\&$\text{length :}$$\frac{b-a}{n}$$\text{ and variable heights :}$f(x_i)$\text{, which depend on the function value at }$x_i\&$\text{In our case, we have function values over an interval:}$\&f(-9)=-17\&f(-2)=11\&f(5)=17\&f(12)=1\&f(19)=-16\&f(26)=13\&$\text{Although the total interval has a length of }$35\&$\text{, notice the points are equidistant, with }$5$\text{ intervals between.}$\&$\text{Each equidistant interval has length: }$7\&$\text{Since we are using the left point of each interval, we disregard the larst function value:}$\&\int_${-9}^{26}$f(x)\approx7[(-17)+(11)+(17)+(1)+(-16)]\&\int_${-9}^{26}$f(x)\approx-28\end{align}$

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Question

$\begin{align}&f(-10)=-19\&f(-3)=7\&f(4)=4\&f(11)=1\&$\text{Approximate, using a left Riemann sum, }$\int_${-10}^{11}$f(x)\end{align}$

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Answer

$\begin{align}&$\text{A Riemann sum integral approximation over an interval [a, b]}$\&$\text{with n subintervals follows the form:}$\&\int_$a^b$ f(x)dx \approx $\frac{b-a}{n}$(f(x_1)+f(x_2)+...+f(x_n))\&$\text{It is essentially a sum of n rectangles, each with a base of}$\&$\text{length :}$$\frac{b-a}{n}$$\text{ and variable heights :}$f(x_i)$\text{, which depend on the function value at }$x_i\&$\text{In our case, we have function values over an interval:}$\&f(-10)=-19\&f(-3)=7\&f(4)=4\&f(11)=1\&$\text{Although the total interval has a length of }$21\&$\text{Notice the points are equidistant, with }$3$\text{ intervals between.}$\&$\text{Each equidistant interval has length: }$7\&$\text{Since we are using the left point of each interval, we disregard the larst function value:}$\&\int_${-10}^{11}$f(x)\approx7[(-19)+(7)+(4)]\&\int_${-10}^{11}$f(x)\approx-56\end{align}$

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Question

$\begin{align}&$\text{Approximate }$\int_${-6}^{34}$f(x)\&$\text{Using a left Riemann sum, if }$f(x)$\text{ has the values:}$\&f(-6)=8\&f(2)=2\&f(10)=-4\&f(18)=-18\&f(26)=11\&f(34)=-7\end{align}$

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Answer

$\begin{align}&$\text{A Riemann sum integral approximation over an interval [a, b]}$\&$\text{with n subintervals follows the form:}$\&\int_$a^b$ f(x)dx \approx $\frac{b-a}{n}$(f(x_1)+f(x_2)+...+f(x_n))\&$\text{It is essentially a sum of n rectangles, each with a base of}$\&$\text{length :}$$\frac{b-a}{n}$$\text{ and variable heights :}$f(x_i)$\text{, which depend on the function value at }$x_i\&$\text{In our case, we have function values over an interval:}$\&f(-6)=8\&f(2)=2\&f(10)=-4\&f(18)=-18\&f(26)=11\&f(34)=-7\&$\text{Although the total interval has a length of }$40\&$\text{Notice the points are equidistant, with }$5$\text{ intervals between.}$\&$\text{Each equidistant interval has length: }$8\&$\text{Since we are using the left point of each interval, we disregard the larst function value:}$\&\int_${-6}^{34}$f(x)\approx8[(8)+(2)+(-4)+(-18)+(11)]\&\int_${-6}^{34}$f(x)\approx-8\end{align}$

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Question

$\begin{align}&$\text{The function }$f(x)$\text{ has the following values on the interval }$x=[-8,2]:\&f(-8)=2\&f(-6)=-1\&f(-4)=16\&f(-2)=12\&f(0)=10\&f(2)=-18\&$\text{Approximate the integral of }$f(x)$\text{ on this interval using left Riemann sums.}$\end{align}$

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Answer

$\begin{align}&$\text{A Riemann sum integral approximation over an interval [a, b]}$\&$\text{with n subintervals follows the form:}$\&\int_$a^b$ f(x)dx \approx $\frac{b-a}{n}$(f(x_1)+f(x_2)+...+f(x_n))\&$\text{It is essentially a sum of n rectangles, each with a base of}$\&$\text{length :}$$\frac{b-a}{n}$$\text{ and variable heights :}$f(x_i)$\text{, which depend on the function value at }$x_i\&$\text{In our case, we have function values over an interval:}$\&f(-8)=2\&f(-6)=-1\&f(-4)=16\&f(-2)=12\&f(0)=10\&f(2)=-18\&$\text{Although the total interval has a length of }$10\&$\text{Notice the points are equidistant, with }$5$\text{ intervals between.}$\&$\text{Each equidistant interval has length: }$2\&$\text{Since we are using the left point of each interval, we disregard the larst function value:}$\&\int_${-8}^{2}$f(x)\approx2[(2)+(-1)+(16)+(12)+(10)]\&\int_${-8}^{2}$f(x)\approx78\end{align}$

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Question

$\begin{align}&$\text{The function }$f(x)$\text{ has the following values on the interval }$x=[5,29]:\&f(5)=8\&f(13)=-15\&f(21)=9\&f(29)=-16\&$\text{Approximate the integral of }$f(x)$\text{ on this interval using left Riemann sums.}$\end{align}$

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Answer

$\begin{align}&$\text{A Riemann sum integral approximation over an interval [a, b]}$\&$\text{with n subintervals follows the form:}$\&\int_$a^b$ f(x)dx \approx $\frac{b-a}{n}$(f(x_1)+f(x_2)+...+f(x_n))\&$\text{It is essentially a sum of n rectangles, each with a base of}$\&$\text{length :}$$\frac{b-a}{n}$$\text{ and variable heights :}$f(x_i)$\text{, which depend on the function value at }$x_i\&$\text{In our case, we have function values over an interval:}$\&f(5)=8\&f(13)=-15\&f(21)=9\&f(29)=-16\&$\text{Although the total interval has a length of }$24\&$\text{Notice the points are equidistant, with }$3$\text{ intervals between.}$\&$\text{Each equidistant interval has length: }$8\&$\text{Since we are using the left point of each interval, we disregard the larst function value:}$\&\int_${5}^{29}$f(x)\approx8[(8)+(-15)+(9)]\&\int_${5}^{29}$f(x)\approx16\end{align}$

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Question

$\begin{align}&f(0)=10\&f(4)=3\&f(8)=10\&f(12)=-11\&f(16)=10\&$\text{Approximate, using a left Riemann sum, }$\int_${0}^{16}$f(x)\end{align}$

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$\begin{align}&$\text{A Riemann sum integral approximation over an interval [a, b]}$\&$\text{with n subintervals follows the form:}$\&\int_$a^b$ f(x)dx \approx $\frac{b-a}{n}$(f(x_1)+f(x_2)+...+f(x_n))\&$\text{It is essentially a sum of n rectangles, each with a base of}$\&$\text{length :}$$\frac{b-a}{n}$$\text{ and variable heights :}$f(x_i)$\text{, which depend on the function value at }$x_i\&$\text{In our case, we have function values over an interval:}$\&f(0)=10\&f(4)=3\&f(8)=10\&f(12)=-11\&f(16)=10\&$\text{Although the total interval has a length of }$16\&$\text{Notice the points are equidistant, with }$4$\text{ intervals between.}$\&$\text{Each equidistant interval has length: }$4\&$\text{Since we are using the left point of each interval, we disregard the larst function value:}$\&\int_${0}^{16}$f(x)\approx4[(10)+(3)+(10)+(-11)]\&\int_${0}^{16}$f(x)\approx48\end{align}$

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