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What is the value of the integral of x^3 from 0 to 2?

User Sme
by
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2 Answers

4 votes

Answer:

To find the value of the integral of
{x}^(3) from 0 to 2, we need to evaluate the definite integral:


\int_(0)^(2) {x}^(3) dx

Using the power rule of integration, we can integrate the function as follows:


\int {x}^(3) dx = \frac{{x}^(4)}{4} + C

where C is the constant of integration.

Plugging in the limits of integration, we get:


\int_(0)^(2) {x}^(3) dx = \left[\frac{{x}^(4)}{4}\right]_(0)^(2)


= \left(\frac{{2}^(4)}{4}\right) - \left(\frac{{0}^(4)}{4}\right)


= (16)/(4) - 0


= 4

Therefore, the value of the integral of
{x}^(3) from 0 to 2 is 4.

User Jesukumar
by
8.2k points
1 vote

Answer: 4

=======================================

Work Shown:


\displaystyle \int_(0)^(2)\text{x}^3d\text{x}\\\\\\\displaystyle (1)/(4)\text{x}^(4)+C\bigg|_(0)^(2)\\\\\\\displaystyle \left[(1)/(4)2^(4)+C\right]-\left[(1)/(4)0^(4)+C\right]\\\\\\4\\\\\\\text{Therefore,} \ \displaystyle \int_(0)^(2)\text{x}^3d\text{x} = 4\\\\

-----------

The integral rule I used in the 2nd step was


\displaystyle \int \text{x}^(n)d\text{x} = (1)/(n+1)\text{x}^(n+1)+C\\\\

in which you can verify it's valid by differentiating
(1)/(n+1)\text{x}^(n+1)+C\\\\ to end up with
\text{x}^(n) again.

User Yuly
by
8.2k points

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