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Calculus 2 : Integrals

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Example Questions

Example Question #196 : Definite Integrals

Evaluate the following integral:

$\int_{\pi}^{0}Sin(x) dx$

Possible Answers:

$\frac{1}{2}$

Correct answer:

Explanation:

Recall that given two real numbers b and a with b>a the integral $\int_{b}^{a}f(x)dx$ is equivalent to $-\int_{a}^{b}f(x)dx$ .

Therefore,

$\int_{\pi}^{0}Sin(x) dx$

is equivalent

$-\int_{0}^{\pi}Sin(x) dx$ .

Evaluating the integral one finds

$-\int_{0}^{\pi}Sin(x) dx= -(-Cos(\pi)-(-Cos(0))))=-(-(-1)-1))=-2$

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Example Question #197 : Definite Integrals

$\begin{align*}&\text{Evaluate the integral: }\\&\int_{10}^{15}(\frac{x^{2}}{10})dx\end{align*}$

Possible Answers:

15.22

166.25

79.17

9.90

Correct answer:

79.17

Explanation:

$\begin{align*}&\text{In performing an integral, familiarity with derivative rules}\\&\text{can prove useful. After all, an integral is essentially the}\\&\text{opposite operation of a derivative. Consider how in a derivative}\\&\text{we’d use the chain rule to multiply the derivative of the “inside”}\\&\text{of a function by the derivative of the outside? Integrals, in}\\&\text{similar fashion, entail a division.}\\&\text{Utilizing integral rules:}\\&\int x^{a}=\frac{x^{a+1}}{a+1}\\&\int_{10}^{15}(\frac{x^{2}}{10})dx=\frac{x^{3}}{30}|_{10}^{15}\\&\text{Now we just subtract the function value at the lower bound from that of the upper bound:}\\&\int_{10}^{15}(\frac{x^{2}}{10})dx=\frac{15^{3}}{30}-(\frac{10^{3}}{30})\\&\int_{10}^{15}(\frac{x^{2}}{10})dx=79.17\end{align*}$

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Example Question #198 : Definite Integrals

$\begin{align*}&\text{Evaluate the integral: }\\&\int_{-4}^{-2}(\frac{1}{(6\cdot 2^x)})dx\end{align*}$

Possible Answers:

6.64

2.89

0.42

26.26

Correct answer:

2.89

Explanation:

$\begin{align*}&\text{In performing an integral, familiarity with derivative rules}\\&\text{can prove useful. After all, an integral is essentially the}\\&\text{opposite operation of a derivative. Consider how in a derivative}\\&\text{we’d use the chain rule to multiply the derivative of the “inside”}\\&\text{of a function by the derivative of the outside? Integrals, in}\\&\text{similar fashion, entail a division.}\\&\text{Utilizing integral rules:}\\&\int[b^{ax}]=\frac{b^{ax}}{aln(b)}\\&\int_{-4}^{-2}(\frac{1}{(6\cdot 2^x)})dx=-\frac{1}{(6\cdot 2^xln(2))}|_{-4}^{-2}\\&\text{Now we just subtract the function value at the lower bound from that of the upper bound:}\\&\int_{-4}^{-2}(\frac{1}{(6\cdot 2^x)})dx=-\frac{1}{(6\cdot 2^{-2}ln(2))}-(-\frac{1}{(6\cdot 2^{-4}ln(2))})\\&\int_{-4}^{-2}(\frac{1}{(6\cdot 2^x)})dx=2.89\end{align*}$

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Example Question #201 : Definite Integrals

$\begin{align*}&\text{Evaluate the integral: }\\&\int_{-25}^{-20.5}(\frac{4}{(3x)})dx\end{align*}$

Possible Answers:

-0.05

-0.03

-0.53

-0.26

Correct answer:

-0.26

Explanation:

$\begin{align*}&\text{In performing an integral, familiarity with derivative rules}\\&\text{can prove useful. After all, an integral is essentially the}\\&\text{opposite operation of a derivative. Consider how in a derivative}\\&\text{we’d use the chain rule to multiply the derivative of the “inside”}\\&\text{of a function by the derivative of the outside? Integrals, in}\\&\text{similar fashion, entail a division.}\\&\text{Utilizing integral rules:}\\&\int_{-25}^{-20.5}(\frac{4}{(3x)})dx=\frac{(4ln(x))}{3}|_{-25}^{\frac{-41}{2}}\\&\text{Now we just subtract the function value at the lower bound from that of the upper bound:}\\&\int_{-25}^{-20.5}(\frac{4}{(3x)})dx=\frac{(4ln(-\frac{41}{2}))}{3}-(\frac{(4ln(-25))}{3})\\&\int_{-25}^{-20.5}(\frac{4}{(3x)})dx=-0.26\end{align*}$

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Example Question #202 : Definite Integrals

$\begin{align*}&\text{Evaluate the integral: }\\&\int_{-8}^{-6}(\frac{(43e^{(x)})}{5})dx\end{align*}$

Possible Answers:

0.02

0.05

0.15

0.12

Correct answer:

0.02

Explanation:

$\begin{align*}&\text{In performing an integral, familiarity with derivative rules}\\&\text{can prove useful. After all, an integral is essentially the}\\&\text{opposite operation of a derivative. Consider how in a derivative}\\&\text{we’d use the chain rule to multiply the derivative of the “inside”}\\&\text{of a function by the derivative of the outside? Integrals, in}\\&\text{similar fashion, entail a division.}\\&\text{Utilizing integral rules:}\\&\int[e^{ax}]=\frac{e^{ax}}{a}\\&\int_{-8}^{-6}(\frac{(43e^{(x)})}{5})dx=\frac{(43e^{(x)})}{5}|_{-8}^{-6}\\&\text{Now we just subtract the function value at the lower bound from that of the upper bound:}\\&\int_{-8}^{-6}(\frac{(43e^{(x)})}{5})dx=\frac{(43e^{(-6)})}{5}-(\frac{(43e^{(-8)})}{5})\\&\int_{-8}^{-6}(\frac{(43e^{(x)})}{5})dx=0.02\end{align*}$

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Example Question #203 : Definite Integrals

$\begin{align*}&\text{Evaluate the integral: }\\&\int_{18}^{22}(\frac{19}{(6x)})dx\end{align*}$

Possible Answers:

0.06

0.64

1.78

4.45

Correct answer:

0.64

Explanation:

$\begin{align*}&\text{In performing an integral, familiarity with derivative rules}\\&\text{can prove useful. After all, an integral is essentially the}\\&\text{opposite operation of a derivative. Consider how in a derivative}\\&\text{we’d use the chain rule to multiply the derivative of the “inside”}\\&\text{of a function by the derivative of the outside? Integrals, in}\\&\text{similar fashion, entail a division.}\\&\text{Utilizing integral rules:}\\&\int_{18}^{22}(\frac{19}{(6x)})dx=\frac{(19ln(x))}{6}|_{18}^{22}\\&\text{Now we just subtract the function value at the lower bound from that of the upper bound:}\\&\int_{18}^{22}(\frac{19}{(6x)})dx=\frac{(19ln(22))}{6}-(\frac{(19ln(18))}{6})\\&\int_{18}^{22}(\frac{19}{(6x)})dx=0.64\end{align*}$

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Example Question #204 : Definite Integrals

$\begin{align*}&\text{Evaluate the integral: }\\&\int_{-3.5}^{0.5}(\frac{(5e^{(-x)})}{9})dx\end{align*}$

Possible Answers:

2.96

2.05

45.15

18.06

Correct answer:

18.06

Explanation:

$\begin{align*}&\text{In performing an integral, familiarity with derivative rules}\\&\text{can prove useful. After all, an integral is essentially the}\\&\text{opposite operation of a derivative. Consider how in a derivative}\\&\text{we’d use the chain rule to multiply the derivative of the “inside”}\\&\text{of a function by the derivative of the outside? Integrals, in}\\&\text{similar fashion, entail a division.}\\&\text{Utilizing integral rules:}\\&\int[e^{ax}]=\frac{e^{ax}}{a}\\&\int_{-3.5}^{0.5}(\frac{(5e^{(-x)})}{9})dx=-\frac{(5e^{(-x)})}{9}|_{-3.5}^{0.5}\\&\text{Now we just subtract the function value at the lower bound from that of the upper bound:}\\&\int_{-3.5}^{0.5}(\frac{(5e^{(-x)})}{9})dx=-\frac{(5e^{(-0.5)})}{9}-(-\frac{(5e^{(--3.5)})}{9})\\&\int_{-3.5}^{0.5}(\frac{(5e^{(-x)})}{9})dx=18.06\end{align*}$

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Example Question #205 : Definite Integrals

$\begin{align*}&\text{Evaluate the integral: }\\&\int_{-26}^{-21}(\frac{25}{(9x)})dx\end{align*}$

Possible Answers:

-1.36

-5.58

-0.10

-0.59

Correct answer:

-0.59

Explanation:

$\begin{align*}&\text{In performing an integral, familiarity with derivative rules}\\&\text{can prove useful. After all, an integral is essentially the}\\&\text{opposite operation of a derivative. Consider how in a derivative}\\&\text{we’d use the chain rule to multiply the derivative of the “inside”}\\&\text{of a function by the derivative of the outside? Integrals, in}\\&\text{similar fashion, entail a division.}\\&\text{Utilizing integral rules:}\\&\int_{-26}^{-21}(\frac{25}{(9x)})dx=\frac{(25ln(x))}{9}|_{-26}^{-21}\\&\text{Now we just subtract the function value at the lower bound from that of the upper bound:}\\&\int_{-26}^{-21}(\frac{25}{(9x)})dx=\frac{(25ln(-21))}{9}-(\frac{(25ln(-26))}{9})\\&\int_{-26}^{-21}(\frac{25}{(9x)})dx=-0.59\end{align*}$

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Example Question #206 : Definite Integrals

$\begin{align*}&\text{Evaluate the integral: }\\&\int_{-26}^{-\frac{45}{2}}(\frac{(7cos(4x))}{37})dx\end{align*}$

Possible Answers:

-0.01

-0.52

-0.21

-0.06

Correct answer:

-0.06

Explanation:

$\begin{align*}&\text{In performing an integral, familiarity with derivative rules}\\&\text{can prove useful. After all, an integral is essentially the}\\&\text{opposite operation of a derivative. Consider how in a derivative}\\&\text{we’d use the chain rule to multiply the derivative of the “inside”}\\&\text{of a function by the derivative of the outside? Integrals, in}\\&\text{similar fashion, entail a division.}\\&\text{Utilizing integral rules:}\\&\int[cos(ax)]=\frac{sin(ax)}{a}\\&\int_{-26}^{-\frac{45}{2}}(\frac{(7cos(4x))}{37})dx=\frac{(7sin(4x))}{148}|_{-26}^{-\frac{45}{2}}\\&\text{Now we just subtract the function value at the lower bound from that of the upper bound:}\\&\int_{-26}^{-\frac{45}{2}}(\frac{(7cos(4x))}{37})dx=\frac{(7sin(4(-\frac{45}{2})))}{148}-(\frac{(7sin(4(-26)))}{148})\\&\int_{-26}^{-\frac{45}{2}}(\frac{(7cos(4x))}{37})dx=-0.06\end{align*}$

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Example Question #207 : Definite Integrals

$\begin{align*}&\text{Evaluate the integral: }\\&\int_{-12}^{-8.5}(\frac{(21sin(x + 1))}{4})dx\end{align*}$

Possible Answers:

-0.21

-1.80

-0.26

-0.82

Correct answer:

-1.80

Explanation:

$\begin{align*}&\text{In performing an integral, familiarity with derivative rules}\\&\text{can prove useful. After all, an integral is essentially the}\\&\text{opposite operation of a derivative. Consider how in a derivative}\\&\text{we’d use the chain rule to multiply the derivative of the “inside”}\\&\text{of a function by the derivative of the outside? Integrals, in}\\&\text{similar fashion, entail a division.}\\&\text{Utilizing integral rules:}\\&\int[sin(ax)]=-\frac{cos(ax)}{a}\\&\int_{-12}^{-8.5}(\frac{(21sin(x + 1))}{4})dx=-\frac{(21cos(x + 1))}{4}|_{-12}^{-8.5}\\&\text{Now we just subtract the function value at the lower bound from that of the upper bound:}\\&\int_{-12}^{-8.5}(\frac{(21sin(x + 1))}{4})dx=-\frac{(21cos((-8.5) + 1))}{4}-(-\frac{(21cos((-12) + 1))}{4})\\&\int_{-12}^{-8.5}(\frac{(21sin(x + 1))}{4})dx=-1.80\end{align*}$

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Calculus 2 : Integrals

Study concepts, example questions & explanations for Calculus 2

All Calculus 2 Resources

Example Questions

Example Question #196 : Definite Integrals

Example Question #197 : Definite Integrals

Example Question #198 : Definite Integrals

Example Question #201 : Definite Integrals

Example Question #202 : Definite Integrals

Example Question #203 : Definite Integrals

Example Question #204 : Definite Integrals

Example Question #205 : Definite Integrals

Example Question #206 : Definite Integrals

Example Question #207 : Definite Integrals

All Calculus 2 Resources

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