AskSia

Plus

Compute the area between the curves

y = x^2

and

y = 2x

Nov 11, 2023

Compute the area between the curves

y = x^2

and

y = 2x

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

To compute the area between the curves

y = x^2

and

y = 2x

, we first need to find the points of intersection of the two curves. These points will provide the limits of integration for calculating the area.

#step 2#

Setting the two equations equal to each other to find the points of intersection:

x^2 = 2x

Solving this equation gives us the x-values of the intersection points.

#step 3#

Factoring the equation from step 2, we get:

x(x - 2) = 0

This gives us two solutions:

x = 0

and

x = 2

. These are the x-coordinates of the points where the two curves intersect.

#step 4#

Now that we have the points of intersection, we can set up the integral to find the area between the curves. The area

A

is given by the integral of the top function minus the bottom function from the leftmost to the rightmost intersection points.

#step 5#

The curve

y = 2x

is above

y = x^2

between

x = 0

and

x = 2

. Thus, the area

A

is given by:

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

#step 6#

Evaluating the integral, we have:

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

#step 7#

Plugging in the limits of integration, we get:

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

Simplifying this expression will give us the area between the curves.

#step 8#

After simplification, the area is:

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

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

Explanation

In this case, we found the points of intersection to be at

x = 0

and

x = 2

. We then set up an integral with these limits and integrated the difference of the two functions,

2x - x^2

, to find the area between the curves. The final answer is the definite integral evaluated from

x = 0

x = 2