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Compute the area of the region enclosed by the parabola

y = x^2

and the line $...

Dec 22, 2023

Compute the area of the region enclosed by the parabola

y = x^2

and the line

y = 4x - 4

Solution by Steps

step 1

Find the points of intersection between the parabola

y = x^2

and the line

y = 4x - 4

by setting them equal to each other:

x^2 = 4x - 4

step 2

Solve the quadratic equation

x^2 - 4x + 4 = 0

to find the intersection points. This can be factored as

(x - 2)^2 = 0

, giving us

x = 2

as the only solution. Thus, the parabola and the line intersect at the point

(2, 4)

step 3

Set up the integral to compute the area between the parabola and the line from

x = 0

x = 2

\text{Area} = \int_{0}^{2} (4x - 4 - x^2) \, dx

step 4

Evaluate the integral:

\text{Area} = \left[ 2x^2 - 4x - \frac{x^3}{3} \right]_{0}^{2}

step 5

Plug in the limits of integration:

\text{Area} = \left( 2(2)^2 - 4(2) - \frac{(2)^3}{3} \right) - \left( 2(0)^2 - 4(0) - \frac{(0)^3}{3} \right)

step 6

Simplify to find the area:

\text{Area} = \left( 8 - 8 - \frac{8}{3} \right) - (0) = -\frac{8}{3}

Since area cannot be negative, we take the absolute value:

\text{Area} = \frac{8}{3}

Answer

The area of the region enclosed by the parabola

y = x^2

and the line

y = 4x - 4

\frac{8}{3}

square units.

Key Concept

Integration to find the area between curves

Explanation

The area between two curves is found by integrating the difference of the functions (top function minus bottom function) over the interval defined by their points of intersection.