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Compute the area between the curves

y = x^2

and

y = 2x

Mar 2, 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 from the left intersection point to the right intersection point

step 2

First, we find the intersection points by setting

x^2 = 2x

. Solving this, we get

x = 0

and

x = 2

step 3

The area between the curves is given by the integral

\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 definite integral, we find

\left[ 2^2 - \frac{2^3}{3} \right] - \left[ 0^2 - \frac{0^3}{3} \right] = 4 - \frac{8}{3} = \frac{12}{3} - \frac{8}{3} = \frac{4}{3}

Answer

\frac{4}{3}

or approximately 1.3333

Key Concept

Definite Integral to Find Area Between Curves

Explanation

The area between two curves from a to b is found by integrating the difference of the functions (top function minus bottom function) from a to b.