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Compute the area between the curves

y = x^2

and

y = 2x

Dec 12, 2023

Compute the area between the curves

y = x^2

and

y = 2x

Solution by Steps

step 1

Find the points of intersection of the curves

y = x^2

and

y = 2x

by setting them equal to each other:

x^2 = 2x.

step 2

Solve the equation from step 1 for

x

x^2 - 2x = 0,

x(x - 2) = 0,

which gives us

x = 0

and

x = 2

step 3

Set up the integral to find the area between the curves from

x = 0

x = 2

\text{Area} = \int_{0}^{2} (2x - x^2) \, dx.

step 4

Evaluate the integral using the power rule for integration:

\text{Area} = \left[ x^2 - \frac{x^3}{3} \right]_{0}^{2}.

step 5

Plug in the limits of integration:

\text{Area} = \left[ 2^2 - \frac{2^3}{3} \right] - \left[ 0^2 - \frac{0^3}{3} \right],

\text{Area} = \left[ 4 - \frac{8}{3} \right] - [0],

\text{Area} = 4 - \frac{8}{3},

\text{Area} = \frac{12}{3} - \frac{8}{3},

\text{Area} = \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.