Integrate 41/3x^2*-1 from 0 to 3

\int_{0}^{3}\left(4-\frac{1}{3} x^2\right)dx

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Answer

$9$

Step by step solution

Problem

$\int_{0}^{3}\left(4-\frac{1}{3} x^2\right)dx$
1

Multiply $-1$ times $\frac{1}{3}$

$\int_{0}^{3}\left(4-\frac{1}{3}x^2\right)dx$
2

The integral of a sum of two or more functions is equal to the sum of their integrals

$\int_{0}^{3}-\frac{1}{3}x^2dx+\int_{0}^{3}4dx$
3

The integral of a constant is equal to the constant times the integral's variable

$\int_{0}^{3}-\frac{1}{3}x^2dx+\left[4x\right]_{0}^{3}$
4

Evaluate the definite integral

$\int_{0}^{3}-\frac{1}{3}x^2dx-1\cdot 0\cdot 4+3\cdot 4$
5

Any expression multiplied by $0$ is equal to $0$

$\int_{0}^{3}-\frac{1}{3}x^2dx+0+3\cdot 4$
6

Multiply $4$ times $3$

$\int_{0}^{3}-\frac{1}{3}x^2dx+0+12$
7

Add the values $12$ and $0$

$\int_{0}^{3}-\frac{1}{3}x^2dx+12$
8

Taking the constant out of the integral

$12-\frac{1}{3}\int_{0}^{3} x^2dx$
9

Apply the power rule for integration, $\displaystyle\int x^n dx=\frac{x^{n+1}}{n+1}$, where $n$ represents a constant function

$\left[-\frac{1}{3}\cdot\frac{x^{3}}{3}\right]_{0}^{3}+12$
10

Simplify the fraction

$\left[-\frac{1}{9}x^{3}\right]_{0}^{3}+12$
11

Evaluate the definite integral

$12-1\cdot 0^{3}\left(-0.1111\right)+3^{3}\left(-0.1111\right)$
12

Multiply $-\frac{1}{9}$ times $-1$

$12+0^{3}\cdot 0.1111+3^{3}\left(-0.1111\right)$
13

Calculate the power

$12+0\cdot 0.1111+27\left(-0.1111\right)$
14

Any expression multiplied by $0$ is equal to $0$

$12+0+27\left(-0.1111\right)$
15

Add the values $0$ and $12$

$27\left(-0.1111\right)+12$
16

Multiply $-\frac{1}{9}$ times $27$

$12-3$
17

Subtract the values $12$ and $-3$

$9$

Answer

$9$

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Problem Analysis

Main topic:

Integral calculus

Time to solve it:

0.22 seconds

Views:

127