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Find the area of the region bounded by

y = x^2 - 2x

and

y = -x

Apr 28, 2024

Find the area of the region bounded by

y = x^2 - 2x

and

y = -x

Solution by Steps

step 1

Find the points of intersection of the curves

y = x^2 - 2x

and

y = -x

step 2

Set

x^2 - 2x = -x

and solve for

x

step 3

Adding

x

to both sides gives

x^2 - x = 0

step 4

Factor out

x

to get

x(x - 1) = 0

step 5

Set each factor equal to zero:

x = 0

and

x - 1 = 0

step 6

Solve for

x

to get the intersection points:

x = 0

and

x = 1

step 7

Integrate the function

x^2 - 2x - (-x) = x^2 - x

from

x = 0

x = 1

step 8

The integral of

x^2 - x

\frac{1}{3}x^3 - \frac{1}{2}x^2

step 9

Evaluate the integral from

x = 0

x = 1

to find the area

step 10

Plug in the upper limit:

\frac{1}{3}(1)^3 - \frac{1}{2}(1)^2 = \frac{1}{3} - \frac{1}{2}

step 11

Plug in the lower limit:

\frac{1}{3}(0)^3 - \frac{1}{2}(0)^2 = 0

step 12

Subtract the lower limit result from the upper limit result to get the area:

\left(\frac{1}{3} - \frac{1}{2}\right) - 0 = \frac{1}{3} - \frac{1}{2}

step 13

Simplify the difference to get the final area:

\frac{1}{3} - \frac{1}{2} = \frac{2}{6} - \frac{3}{6} = -\frac{1}{6}

step 14

Since area cannot be negative, take the absolute value:

|\frac{-1}{6}| = \frac{1}{6}

Answer

\frac{1}{6}

Key Concept

Integration to find area between 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. The absolute value is taken since area is always positive.