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Compute the area between the curves

y = x^2

and

y = 2x

May 9, 2024

Compute the area between the curves

y = x^2

and

y = 2x

Generated Graph

Solution by Steps

step 1

To find the area between the curves

y = x^2

and

y = 2x

, we need to integrate the difference of the functions

step 2

Set up the integral for the area between the curves from

x = 0

x = 2

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

step 3

Compute the integral:

\int (2x - x^2) dx = x^2 - \frac{x^3}{3} + C

step 4

Evaluate the integral from

x = 0

x = 2

\left[ x^2 - \frac{x^3}{3} \right]_0^2 = \left[ 2^2 - \frac{2^3}{3} \right] - \left[ 0^2 - \frac{0^3}{3} \right]

step 5

Simplify the result:

4 - \frac{8}{3} = \frac{12}{3} - \frac{8}{3} = \frac{4}{3}

Answer

The area between the curves

y = x^2

and

y = 2x

from

x = 0

x = 2

\frac{4}{3}

or approximately 1.3333.

Key Concept

Area between two curves

Explanation

The area between two curves is found by integrating the difference of the functions over the interval of intersection.