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Calculus 2 : Definite Integrals

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Example Questions

Example Question #101 : Definite Integrals

$\int_{0}^{2}9x^2-3x+1dx$

Possible Answers:

Correct answer:

Explanation:

First, integrate the expression. Remember to raise the exponent by 1 and then put that result on the denominator:

$9(\frac{x^3}{3})-\frac{3x^2}{2}+x=3x^3-\frac{3x^2}{2}+x$ .

Then, evaluate at 2 and then 0.

Subtract the two:

3(2^3)-3(2)+2=24-6+2=20.

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Example Question #102 : Definite Integrals

What is the area underneath the curve f(x)=x^2-3x+1 from x=4 to x=6 and bounded by the axis?

Possible Answers:

$\frac{68}{5}$

$\frac{67}{3}$

$\frac{68}{3}$

Correct answer:

$\frac{68}{3}$

Explanation:

First set up the integral expression:

$\int_{4}^{6}x^2-3x+1dx$ .

Then, integrate each term separately. Remember to raise the exponent by 1 and then put that result on the denominator as well:

$\frac{x^3}{3}-\frac{3x^2}{2}+x$ .

Then evaluate at 6 and then 4. Subtract the results.

$(\frac{216}{3}-\frac{108}{2}+6)-(\frac{64}{3}-\frac{48}{2}+4)$ .

Simplify to get your answer:

$(72-54+6)-(\frac{64}{3}-24+4)=24-(\frac{64}{3}-20)=\frac{68}{3}$ .

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Example Question #103 : Definite Integrals

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

Possible Answers:

3.0794

2.0578

1.0794

Correct answer:

3.0794

Explanation:

First, chop up the fraction into two separate terms:

$\int \frac{3}{x}+1dx$ .

Then, integrate those to get

$3ln\left | x \right |+x$ .

Then evaluate at 2 and then 1. Subtract the results:

(3ln2+2)-(3ln1+1)=4.0794-1=3.0794 .

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Example Question #104 : Definite Integrals

Calculate the value of the definite integral
$\int_0^{\pi/4} sec^2x\, dx$

Possible Answers:

$-\pi$

Correct answer:

Explanation:

The antiderivative of sec^2x is $tan\,x$ .

Using the corollary to the Fundamental Theorem of Calculus we have

$\int_0^{\pi/4} sec^2x\, dx=tan\,x|_0^{\pi/4}=tan(\pi/4)-tan(0)=1-0=1$

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Example Question #105 : Definite Integrals

Calculate the value of the definite integral
$\int_4^7 \frac{2x}{1+x^2} \,dx$

Possible Answers:

$\frac{2}{3}$

$ln(\frac{7}{4})$

$=ln(\frac{50}{17})$

$\frac{7}{4}$

Correct answer:

$=ln(\frac{50}{17})$

Explanation:

To solve this definite integral, we use u-substitution.

u=1+x^2 and $du=2x\,dx$

We then solve the integral

$\int_{17}^{50} \frac{du}{u}=(ln|u||_{17}^{50}$

=ln50-ln17

$=ln(\frac{50}{17})$

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Example Question #106 : Definite Integrals

Evaluate.

$\int_{20}^{21} \frac{1}{15x- 250} \ dx$

Possible Answers:

0.06

0.04

Answer not listed.

0.02

0.12

Correct answer:

0.02

Explanation:

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

In this case, $f(x) = \frac{1}{15x- 250}$ .

The antiderivative is $F(x) =\frac{\ln(15x-250)}{15}$ .

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

$\int_{20}^{21} \frac{1}{15x- 250} \ dx = \left ( \frac{\ln(15x-250)}{15} \right )_{20}^{21}$

$= \frac{\ln(15(21)-250)}{15} - \left ( \frac{\ln(15(20)-250)}{15} \right )$

=0.02

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Example Question #107 : Definite Integrals

Evaluate.

$\int_{2}^{3}\sin (3x - 7) \ dx$

Possible Answers:

0.32

Answer not listed.

1.98

-1.24

0.53

Correct answer:

0.32

Explanation:

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

In this case, $f(x) = \sin (3x - 7)$ .

The antiderivative is $F(x) =-\frac{\cos(3x-7)}{3}$ .

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

$\int_{2}^{3}\sin (3x - 7) \ dx = \left ( -\frac{\cos(3x-7)}{3} \right )_{2}^{3}$

$= -\frac{\cos(3(3)-7)}{3} - \left ( -\frac{\cos(3(2)-7)}{3} \right )$

=0.32

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Example Question #108 : Definite Integrals

Evaluate.

$\int_{1}^{2} 6x +4 \ dx$

Possible Answers:

Answer not listed.

Correct answer:

Explanation:

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

In this case, f(x) = 6x +4 .

The antiderivative is F(x) =3x^2 + 4x .

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

$\int_{1}^{2} 6x +4 \ dx = \left ( 3x^2 + 4x \right )_{1}^{2}$

$= 3(2)^2 + 4(2) - \left ( 3(1)^2 + 4(1) \right )$

=13

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Example Question #109 : Definite Integrals

Evaluate.

$\int_{3}^{4} \frac{1}{3x-7} \ dx$

Possible Answers:

Answer not listed.

0.31

1.54

0.14

3.69

Correct answer:

0.31

Explanation:

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

In this case, $f(x) = \frac{1}{3x-7}$ .

The antiderivative is $F(x) =\frac{\ln(3x-7)}{3}$ .

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

$\int_{3}^{4} \frac{1}{3x-7} \ dx= \left ( \frac{\ln(3x-7)}{3} \right )_{3}^{4}$

$= \frac{\ln(3(4)-7)}{3} - \left ( \frac{\ln(3(3)-7)}{3} \right )$

=0.31

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Example Question #110 : Definite Integrals

Evaluate.

$\int_{0}^{2} x^6 \ dx$

Possible Answers:

$\frac{1}{3}$

$\frac{8}{3}$

$\frac{128}{7}$

$\frac{22}{3}$

Answer not listed.

Correct answer:

$\frac{128}{7}$

Explanation:

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

In this case, f(x) = x^6 .

The antiderivative is $F(x) = \frac{x^7}{7}$ .

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

$\int_{0}^{2} x^6 \ dx = \left ( \frac{x^7}{7} \right )_{0}^{2}$

$= \frac{2^7}{7} - \left ( \frac{0^5}{7} \right )$

$= \frac{128}{7}$

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Calculus 2 : Definite Integrals

Study concepts, example questions & explanations for Calculus 2

All Calculus 2 Resources

Example Questions

Example Question #101 : Definite Integrals

Example Question #102 : Definite Integrals

Example Question #103 : Definite Integrals

Example Question #104 : Definite Integrals

Example Question #105 : Definite Integrals

Example Question #106 : Definite Integrals

Example Question #107 : Definite Integrals

Example Question #108 : Definite Integrals

Example Question #109 : Definite Integrals

Example Question #110 : Definite Integrals

All Calculus 2 Resources

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