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Compute the area between the curves

y = x^2

and

y = 2x

Nov 22, 2023

Compute the area between the curves

y = x^2

and

y = 2x

Here is a Step-by-Step solution: #step 1#

To find the area between the curves

y = x^2

and

y = 2x

, we first need to determine the points of intersection of the two curves. This is done by setting the two equations equal to each other and solving for

x

#step 2#

Setting the equations equal to each other:

x^2 = 2x

Solving for

x

gives us the points of intersection:

x^2 - 2x = 0

x(x - 2) = 0

So,

x = 0

x = 2

. These are the points where the curves intersect.

#step 3#

Now, we need to set up the integral to compute the area between the curves from

x = 0

x = 2

. The area

A

is given by the integral of the top function minus the bottom function:

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

#step 4#

Calculating the integral:

A = \int_{0}^{2} (2x - x^2) dx = \left[ x^2 - \frac{x^3}{3} \right]_0^2

#step 5#

Evaluating the integral from

x = 0

x = 2

A = \left[ (2)^2 - \frac{(2)^3}{3} \right] - \left[ (0)^2 - \frac{(0)^3}{3} \right]

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

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

A = \frac{4}{3}

Answer

The area between the curves

y = x^2

and

y = 2x

\frac{4}{3}

square units.

Key Concept

The area between two curves can be found by integrating the difference of the functions that define the curves over the interval of their intersection points.

Explanation

In this case, the area was found by integrating the difference

(2x - x^2)

from

x = 0

x = 2

, which are the intersection points of the curves

y = x^2

and

y = 2x

. The definite integral gives the total area between the curves over this interval.