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Calculus 2 : Definite Integrals

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Example Questions

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Example Question #41 : Finding Integrals

Evaluate this integral.

$\int_{1}^{2} e^x dx$

Possible Answers:

-6.48

4.67

Answer not listed

2.16

9.43

Correct answer:

4.67

Explanation:

$\int_{a}^{b} f(x) dx$

In order to evaluate this integral, first find the antiderivative of f(x).

In this case, f(x) = e^x .

The antiderivative is F(x) = e^x .

Using the Fundamental Theorem of Calculus, evaluate the integral using the antiderivative:

$\int_{1}^{2} e^x dx = \left (e^x \right ) _{1}^{2}$

$= e^{2} - e^{1}$

= 4.67

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Example Question #42 : Finding Integrals

Evaluate this integral.

$\int_{1}^{2} 16x^3 - \frac{1}{x} dx$

Possible Answers:

59.3

Answer not listed

98.4

12.5

64.8

Correct answer:

59.3

Explanation:

$\int_{a}^{b} f(x) dx$

In order to evaluate this integral, first find the antiderivative of f(x).

In this case, $f(x) = 16x^3 - \frac{1}{x}$ .

The antiderivative is F(x) =4x^4 - ln(x) .

Using the Fundamental Theorem of Calculus, evaluate the integral using the antiderivative:

$\int_{1}^{2} 16x^3 - \frac{1}{x} dx = \left (4x^4 - ln(x) \right ) _{1}^{2}$

$= 4(2)^4 - ln(2) - \left (4(1)^4 - ln(1) \right )$

= 59.3

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Example Question #43 : Finding Integrals

Evaluate this integral.

$\int_{0}^{2} x^2 dx$

Possible Answers:

$\frac{4}{5}$

$\frac{3}{2}$

$\frac{7}{8}$

Answer not listed

$\frac{8}{3}$

Correct answer:

$\frac{8}{3}$

Explanation:

$\int_{a}^{b} f(x) dx$

In order to evaluate this integral, first find the antiderivative of f(x).

In this case, f(x) = x^2 .

The antiderivative is $F(x) = \frac{1}{3}x^3$ .

Using the Fundamental Theorem of Calculus, evaluate the integral using the antiderivative:

$\int_{0}^{2} x^2 dx = \left ( \frac{1}{3}x^3 \right )_{0}^{2}$

$= \frac{1}{3}(2)^3 - \left ( \frac{1}{3}(0)^3 \right )$

$= \frac{8}{3}$

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Example Question #44 : Finding Integrals

Evaluate this integral.

$\int_{0}^{2} x-5} dx$

Possible Answers:

Answer not listed

Correct answer:

Explanation:

$\int_{a}^{b} f(x) dx$

In order to evaluate this integral, first find the antiderivative of f(x).

In this case, f(x) = x-5 .

The antiderivative is $F(x) = \frac{1}{2}x^2 - 5x$ .

Using the Fundamental Theorem of Calculus, evaluate the integral using the antiderivative:

$\int_{0}^{2} x-5 dx = \left ( \frac{1}{2}x^2 - 5x \right )_{0}^{2}$

$= \frac{1}{2}(2)^2 - 5(2) - \left (\frac{1}{2}(0)^2 - 5(0) \right )$

= -8

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Example Question #45 : Finding Integrals

Evaluate this integral.

$\int_{1}^{2} 12x - 1 \ dx$

Possible Answers:

Answer not listed

Correct answer:

Explanation:

$\int_{a}^{b} f(x) dx$

In order to evaluate this integral, first find the antiderivative of f(x).

In this case, f(x) = 12x -1 .

The antiderivative is F(x) = 6x^2 - x .

Using the Fundamental Theorem of Calculus, evaluate the integral using the antiderivative:

$\int_{1}^{2} x-5 \ dx = \left ( 6x^2 - x \right )_{1}^{2}$

$= 6(2)^2 - (2) - \left ( 6(1)^2 - (1)\right )$

= 17

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Example Question #46 : Finding Integrals

Evaluate this integral.

$\int_{0}^{\pi} \sin x \ dx$

Possible Answers:

$\pi$

Answer not listed

Correct answer:

Explanation:

$\int_{a}^{b} f(x) dx$

In order to evaluate this integral, first find the antiderivative of f(x).

In this case, $f(x) = \sin x$ .

The antiderivative is $F(x) = -\cos x$ .

Using the Fundamental Theorem of Calculus, evaluate the integral using the antiderivative:

$\int_{0}^{\pi} \sin x\ dx = \left ( -\cos x \right )_{0}^{\pi}$

$= -\cos (\pi) - \left (-\cos (0) \right )$

= 2

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Example Question #51 : Finding Integrals

Evaluate this integral.

$\int_{2}^{3} 4x \ dx$

Possible Answers:

Answer not listed

Correct answer:

Explanation:

$\int_{a}^{b} f(x) dx$

In order to evaluate this integral, first find the antiderivative of f(x).

In this case, f(x) = 4x .

The antiderivative is F(x) = 2x^2 .

Using the Fundamental Theorem of Calculus, evaluate the integral using the antiderivative:

$\int_{2}^{3} 4x \ dx = \left (2x^2)_{2}^{3}$

$= 2(3)^2 - \left (2(2)^2 \right )$

= 10

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Example Question #52 : Finding Integrals

Evaluate this integral.

$\int_{0}^{\pi} 2 \cos x \ dx$

Possible Answers:

$\pi$

Answer not listed

Correct answer:

Explanation:

$\int_{a}^{b} f(x) dx$

In order to evaluate this integral, first find the antiderivative of f(x).

In this case, $f(x) = \cos x$ .

The antiderivative is $F(x) = \sin x$ .

Using the Fundamental Theorem of Calculus, evaluate the integral using the antiderivative:

$2 \int_{0}^{\pi} \cos x \ dx = \left (\sin x)_{0}^{\pi}$

$= 2 \ \left (\sin(\pi) - \sin (0) \right )$

= 0

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Example Question #53 : Finding Integrals

Evaluate this integral.

$\int_{1}^{2} \frac{3}{x} \ dx$

Possible Answers:

Answer not listed

3.4

2.1

8.7

9.2

Correct answer:

2.1

Explanation:

$\int_{a}^{b} f(x) dx$

In order to evaluate this integral, first find the antiderivative of f(x).

In this case, $f(x) = \frac{1}{x}$ .

The antiderivative is F(x) = ln(x) .

Using the Fundamental Theorem of Calculus, evaluate the integral using the antiderivative:

$3 \int_{1}^{2} \frac{1}{x} dx = (ln (x))_{1}^{2}$

$= 3 \ \left (ln(2) - ln(1) \right )$

= 2.1

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Example Question #54 : Finding Integrals

Evaluate this integral.

$\int_{1}^{2} x^3 + 2 dx$

Possible Answers:

Answer not listed

$\frac{9}{7}$

$\frac{23}{4}$

$\frac{1}{4}$

$\frac{18}{5}$

Correct answer:

$\frac{23}{4}$

Explanation:

$\int_{a}^{b} f(x) dx$

In order to evaluate this integral, first find the antiderivative of f(x).

In this case, f(x) = x^3 + 2 .

The antiderivative is $F(x) = \frac{1}{4}x^4 + 2x$ .

Using the Fundamental Theorem of Calculus, evaluate the integral using the antiderivative:

$\int_{1}^{2} \sin x \ dx = \left (\frac{1}{4} x^4 + 2x \right ) _{1}^{2}$

$= \frac{1}{4} (2)^4 + 2(2) - \left (\frac{1}{4}(1)^4 + 2(1) \right )$

$= \frac{23}{4}$

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Calculus 2 : Definite Integrals

Study concepts, example questions & explanations for Calculus 2

All Calculus 2 Resources

Example Questions

Example Question #41 : Finding Integrals

Example Question #42 : Finding Integrals

Example Question #43 : Finding Integrals

Example Question #44 : Finding Integrals

Example Question #45 : Finding Integrals

Example Question #46 : Finding Integrals

Example Question #51 : Finding Integrals

Example Question #52 : Finding Integrals

Example Question #53 : Finding Integrals

Example Question #54 : Finding Integrals

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