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Compute the area between the curves

y = x^2

and

y = 2x

Apr 1, 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 set up the integral of the difference between the two functions

step 2

The points of intersection are found by setting

x^2 = 2x

, which gives us

x = 0

and

x = 2

. These will be the limits of our integral

step 3

The integral to find the area between the curves is

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

step 4

Computing the integral, we get

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

step 5

Evaluating the integral from 0 to 2, we get

\left[ 2^2 - \frac{2^3}{3} \right] - \left[ 0^2 - \frac{0^3}{3} \right] = 4 - \frac{8}{3}

step 6

Simplifying the result, we find the area to be

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

\frac{4}{3}

square units.

Key Concept

Area between two curves

Explanation

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