Question

Math

Posted 6 months ago

Lit

`Find the area of the region bounded by $y = x^2 - 2x$ and $y = -x$.`

Sign in to unlock the answer

Answer from Sia

Posted 6 months ago

Solution by Steps

step 1

To find the area between the curves $y = x^2 - 2x$ and $y = -x$, we need to set up the integral of the top function minus the bottom function from the leftmost to the rightmost points of intersection

step 2

First, find the points of intersection by setting $x^2 - 2x = -x$. Solving this gives $x^2 - x = 0$, which has solutions $x = 0$ and $x = 1$

step 3

The integral setup is $\int_{0}^{1} ((x^2 - 2x) - (-x)) dx$, which simplifies to $\int_{0}^{1} (x^2 - x) dx$

step 4

Compute the integral $\int_{0}^{1} (x^2 - x) dx$ to find the area

step 5

The antiderivative of $x^2 - x$ is $\frac{x^3}{3} - \frac{x^2}{2}$

step 6

Evaluate the antiderivative from 0 to 1: $\left[\frac{x^3}{3} - \frac{x^2}{2}\right]_{0}^{1} = \left(\frac{1^3}{3} - \frac{1^2}{2}\right) - \left(\frac{0^3}{3} - \frac{0^2}{2}\right)$

step 7

Simplify to get the area: $\frac{1}{3} - \frac{1}{2} = \frac{2}{6} - \frac{3}{6} = -\frac{1}{6}$. Since area cannot be negative, we take the absolute value

Answer

$\frac{1}{6}$

Key Concept

Area between two 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.

Not the question you are looking for? Ask here!

Enter question by text

Enter question by image

Upgrade to Asksia Pro

Join a AskSia's Pro Plan, and get 24/7 AI tutoring for your reviews, assignments, quizzes and exam preps.

Unlimited chat query usages

Strong algorithms that better know you

Early access to new release features

Study Other Question