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What is the integral of $x^2$ from $0$ to $4$

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Evaluate definite integrals using the chain rule

Full solution

Q. What is the integral of

x^2

from

0

to

4

Find Antiderivative: To solve the integral of $x^2$ from $0$ to $4$ , we need to find the antiderivative of $x^2$ and then evaluate it at the upper and lower limits of the integral. $\newline$ The antiderivative of $x^2$ is $(1/3)x^3$ , since the derivative of $(1/3)x^3$ with respect to $x$ is $x^2$ . $\newline$ Calculation: $\int x^2 dx = (1/3)x^3 + C$ , where $0$ $0$ is the constant of integration.
Evaluate at Limits: Now we need to evaluate the antiderivative at the upper limit of $4$ and subtract the evaluation at the lower limit of $0$ .
Calculation: $\left[\frac{1}{3} \times 4^3\right] - \left[\frac{1}{3} \times 0^3\right] = \frac{1}{3} \times 64 - 0 = \frac{64}{3}$ .

More problems from Evaluate definite integrals using the chain rule

Question

Consider the following problem:

\newline

The water level under a bridge is changing at a rate of

r(t)=40 \sin \left(\frac{\pi t}{6}\right)

centimeters per hour (where

t

is the time in hours). At time

t=3

, the water level is

91

centimeters. By how much does the water level change during the

4^{\text {th }}

\newline

Which expression can we use to solve the problem?

\newline

1

\newline

\int_{3}^{4} r(t) d t

\newline

\int_{0}^{4} r(t) d t

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\int_{4}^{5} r(t) d t

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\int_{4}^{4} r(t) d t

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Question

s(t)

gives the number of students enrolled in a school by time

t

\newline

\int_{15}^{18} s^{\prime}(t) d t=20

\newline

1

\newline

(A) The rate of change of enrollment is

20

students per year more in year

18

than it was in year

15

\newline

20

more students enrolled in year

18

15

\newline

(C) Between years

15

18

, the cumulative number of years of schooling of all of the enrolled students is

20

\newline

20

students enrolled in year

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Question

A water tank is filled at a rate of

r(t)

liters per minute (where

t

is the time in minutes).

\newline

\int_{1}^{7} r^{\prime}(t) d t

\newline

1

\newline

(A) The rate at which the tank was filled at

t=7

\newline

B) The average rate of filling between

t=1

t=7

\newline

(C) The change in the rate of filling between

t=1

t=7

\newline

(D) The amount of water filled between

t=1

t=7

Posted 8 months ago

Question

The base of a solid is the region enclosed by the graphs of

y=\sin (x)

y=4-\sqrt{x}

x=2

x=7

\newline

Cross sections of the solid perpendicular to the

x

-axis are rectangles whose height is

2 x

\newline

Which one of the definite integrals gives the volume of the solid?

\newline

1

\newline

\int_{2}^{7}\left[(4-\sqrt{x})^{2}-\sin ^{2}(x)\right] d x

\newline

\int_{2}^{7}[\sin (x)+\sqrt{x}-4] \cdot 2 x d x

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\int_{2}^{7}\left[\sin ^{2}(x)-(4-\sqrt{x})^{2}\right] d x

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\int_{2}^{7}[4-\sqrt{x}-\sin (x)] \cdot 2 x d x

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Question

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\int-4 x^{2} 3^{3 x} d x

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Evaluate the integral.

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\int-4 x^{2} e^{4 x} d x

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\sum_{n=1}^{\infty}\frac{3^{n}n^{2}}{n!}

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Question

f) \lim_{n \to +\infty}\left[\frac{2}{n}+\left(\frac{3}{5}\right)^n\right]

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Question

\int \frac{1}{\cos(x)^{2}}\,dx

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