Approximating Areas With Riemann Sums - AP Calculus AB

1. Use a midpoint Riemann sum with 2 equal subintervals to approximate the value of $\int_0^2 x^3 dx$.
   Answer: 3.5 The interval is $[0, 2]$ with $n=2$ subintervals, so the width is $\Delta x = \frac{2-0}{2} = 1$. The subintervals are $[0,1]$ and $[1,2]$. The midpoints are $0.5$ and $1.5$. The midpoint Riemann sum is $\Delta x [f(0.5) + f(1.5)] = 1[(0.5)^3 + (1.5)^3] = 1[0.125 + 3.375] = 3.5$.
2. Let $f$ be a function such that $f'(x) < 0$ and $f''(x) < 0$ for all $x$ in the interval $[a, b]$. Let $I = \int_a^b f(x) dx$. For a given number of subintervals $n$, which of the following must be true about the left Riemann sum ($L_n$) and trapezoidal sum ($T_n$) approximations?
   Answer: $T_n < I < L_n$ Since $f'(x) < 0$, the function $f$ is decreasing. For a decreasing function, the left Riemann sum $L_n$ is an overestimate of the integral $I$. Since $f''(x) < 0$, the function is concave down. For a concave down function, the trapezoidal sum $T_n$ is an underestimate of the integral $I$. Combining these facts, we have $T_n < I < L_n$.
3. Let $g$ be a function that is strictly decreasing on the interval $[a, b]$. Which of the following statements provides the best comparison between the right Riemann sum approximation ($R_n$) and the true value of the integral $\int_a^b g(x) dx$?
   Answer: $R_n$ is an underestimate of the integral. For a strictly decreasing function on a given subinterval, the function's value at the right endpoint is the minimum value on that subinterval. Therefore, a right Riemann sum, which uses the function values at the right endpoints, will be an underestimate of the true value of the integral.
4. Using the given values, approximate the definite integral $\int_1^7 g(x) dx$ with a right Riemann sum using 3 subintervals of equal width.
   Answer: 70 The interval is $[1, 7]$ with $n=3$ subintervals, so the width of each subinterval is $\Delta x = \frac{7-1}{3} = 2$. The subintervals are $[1,3], [3,5], [5,7]$. For a right Riemann sum, we use the right endpoints $3, 5, 7$. The approximation is $\Delta x [g(3) + g(5) + g(7)] = 2[8 + 12 + 15] = 2[35] = 70$.
5. Use the trapezoidal rule with 4 equal subintervals to approximate $\int_0^2 e^{x^2} dx$.
   Answer: $\frac{1}{4}(e^0 + 2e^{0.25} + 2e^1 + 2e^{2.25} + e^4)$ The interval is $[0, 2]$ with $n=4$ subintervals, so the width is $\Delta x = \frac{2-0}{4} = 0.5$. The endpoints are $0, 0.5, 1, 1.5, 2$. The trapezoidal rule formula is $\frac{\Delta x}{2}[f(x_0) + 2f(x_1) + 2f(x_2) + 2f(x_3) + f(x_4)]$. Plugging in the values: $\frac{0.5}{2}[e^{0^2} + 2e^{0.5^2} + 2e^{1^2} + 2e^{1.5^2} + e^{2^2}]$. This simplifies to $\frac{1}{4}[e^0 + 2e^{0.25} + 2e^1 + 2e^{2.25} + e^4]$.
6. A right Riemann sum with 5 equal subintervals is used to approximate $\int_2^{12} f(x) dx$. Which of the following is the width of each rectangle?
   Answer: 2 The width of each subinterval (rectangle) in a Riemann sum with equal subintervals is given by the formula $\Delta x = \frac{b-a}{n}$, where $[a, b]$ is the interval of integration and $n$ is the number of subintervals. In this case, $a=2, b=12$, and $n=5$. Therefore, the width is $\Delta x = \frac{12-2}{5} = \frac{10}{5} = 2$.
7. A right Riemann sum is used to estimate the distance the runner travels during the first 4 seconds. How does this estimate compare to the actual distance traveled?
   Answer: The estimate is an overestimate because the function is strictly increasing. The distance traveled is the integral of the speed function, $\int_0^4 s(t) dt$. Since the speed function $s(t)$ is strictly increasing, the value at the right endpoint of any subinterval is the maximum value on that subinterval. Therefore, a right Riemann sum will use the maximum value for each rectangle's height, resulting in an overestimate of the actual distance traveled.
8. For a certain continuous function $f(x)$, it is known that a left Riemann sum is always an overestimate and a right Riemann sum is always an underestimate for $\int_a^b f(x) dx$ for any number of subintervals. Which of the following must be true about $f(x)$ on $[a, b]$?
   Answer: $f(x)$ is decreasing If a left Riemann sum is an overestimate, it means the function's value at the left endpoint of each subinterval is the maximum value in that subinterval. This occurs when a function is decreasing. If a right Riemann sum is an underestimate, it means the function's value at the right endpoint is the minimum value, which also occurs when a function is decreasing. Therefore, $f(x)$ must be a decreasing function on the interval.
9. The values of a continuous function $f$ for selected values of $x$ are given as follows: $f(0)=5, f(1)=8, f(2)=13, f(3)=20$. What is the value of a right Riemann sum approximation of $\int_0^3 f(x) dx$ with 3 equal subintervals?
   Answer: 41 The interval of integration is $[0, 3]$ with $n=3$ subintervals, so the width of each subinterval is $\Delta x = \frac{3-0}{3} = 1$. The subintervals are $[0,1], [1,2], [2,3]$. A right Riemann sum uses the right endpoints $x=1, 2, 3$. The approximation is $\Delta x [f(1) + f(2) + f(3)] = 1[8 + 13 + 20] = 41$.
10. Let $f$ be a function that is strictly increasing on the interval $[a, b]$. Which of the following statements must be true about the approximations for $\int_a^b f(x) dx$ using a left Riemann sum ($L_n$) and a right Riemann sum ($R_n$) with $n$ subintervals?
    Answer: $L_n < \int_a^b f(x) dx < R_n$ For a strictly increasing function on a given subinterval, the minimum value occurs at the left endpoint and the maximum value occurs at the right endpoint. Therefore, a left Riemann sum ($L_n$) will be an underestimate of the true integral, and a right Riemann sum ($R_n$) will be an overestimate. This leads to the inequality $L_n < \int_a^b f(x) dx < R_n$.
11. Use a left Riemann sum with 4 equal subintervals to approximate the area of the region bounded by the graph of $f(x) = x^2$, the x-axis, from $x=0$ to $x=4$.
    Answer: 14 The interval is $[0, 4]$ with $n=4$ subintervals, so the width of each subinterval is $\Delta x = \frac{4-0}{4} = 1$. The subintervals are $[0,1], [1,2], [2,3], [3,4]$. For a left Riemann sum, we use the left endpoints of these subintervals, which are $0, 1, 2, 3$. The approximation is $\Delta x [f(0) + f(1) + f(2) + f(3)] = 1[0^2 + 1^2 + 2^2 + 3^2] = 1[0+1+4+9] = 14$.
12. The approximation for $\int_a^b f(x) dx$ using a left Riemann sum is $A_L$ and using a right Riemann sum is $A_R$. If the trapezoidal approximation is $A_T$, which of the following gives an expression for $A_T$ in terms of $A_L$ and $A_R$, assuming equal subintervals?
    Answer: $\frac{A_L + A_R}{2}$ For equal subintervals of width $\Delta x$, the left Riemann sum is $A_L = \Delta x \sum_{i=0}^{n-1} f(x_i)$ and the right Riemann sum is $A_R = \Delta x \sum_{i=1}^{n} f(x_i)$. The trapezoidal sum is $A_T = \frac{\Delta x}{2} \sum_{i=1}^{n} (f(x_{i-1}) + f(x_i)) = \frac{\Delta x}{2} [ (f(x_0)+f(x_1)) + ... + (f(x_{n-1})+f(x_n)) ] = \frac{\Delta x}{2} [f(x_0) + 2f(x_1) + ... + 2f(x_{n-1}) + f(x_n)] = \frac{1}{2} [\Delta x(f(x_0)+...+f(x_{n-1})) + \Delta x(f(x_1)+...+f(x_n))] = \frac{A_L+A_R}{2}$.
13. Using a left Riemann sum with three subintervals of equal width, what is the approximation of the total volume of water that flowed into the reservoir during the 6-hour period?
    Answer: 400 cubic meters The total volume is approximated by the integral of the rate, $\int_0^6 R(t) dt$. The interval is $[0, 6]$ with 3 subintervals, so the width is $\Delta t = \frac{6-0}{3}=2$ hours. A left Riemann sum uses the left endpoints $t=0, 2, 4$. The approximation is $\Delta t [R(0) + R(2) + R(4)] = 2[50+70+80] = 2[200] = 400$. The units are (hours) * (cubic meters/hour) = cubic meters.
14. The function $h(x)$ is twice differentiable and $h''(x) > 0$ for all $x$ in the interval $[0, 4]$. Let $T_4$ be the trapezoidal sum approximation with 4 equal subintervals for $\int_0^4 h(x) dx$. Which statement about $T_4$ must be true?
    Answer: $T_4$ is an overestimate of $\int_0^4 h(x) dx$. The condition $h''(x) > 0$ means that the graph of $h(x)$ is concave up. For a concave up function, the secant line segment connecting the endpoints of any subinterval lies above the curve. Since the trapezoidal rule uses these secant line segments to form the tops of the trapezoids, the area of each trapezoid is greater than the area under the curve on that subinterval. Thus, the trapezoidal sum $T_4$ is an overestimate.
15. The function $f$ is continuous on $[0, 6]$. Using three subintervals of equal width and right endpoints, the right Riemann sum approximation for $\int_0^6 f(x) dx$ is calculated. Which of the following expressions represents this approximation?
    Answer: $2(f(2)+f(4)+f(6))$ The interval is $[0, 6]$ with $n=3$ subintervals of equal width. The width of each subinterval is $\Delta x = \frac{6-0}{3} = 2$. The subintervals are $[0,2], [2,4], [4,6]$. For a right Riemann sum, we use the function values at the right endpoints of these subintervals, which are $x=2, x=4, x=6$. The approximation is given by the sum of the areas of the three rectangles: $f(2) \cdot \Delta x + f(4) \cdot \Delta x + f(6) \cdot \Delta x = \Delta x (f(2)+f(4)+f(6)) = 2(f(2)+f(4)+f(6))$.

Start this quiz