AskSia

Plus

Determine the area under the curve

y = 4 - x^2

from

x = -2

x = 2

Nov 16, 2023

Determine the area under the curve

y = 4 - x^2

from

x = -2

x = 2

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

To determine the area under the curve

y = 4 - x^2

from

x = -2

x = 2

, we will set up the integral from the lower bound of

x = -2

to the upper bound of

x = 2

#step 2#

The integral that represents the area under the curve is given by:

\int_{-2}^{2} (4 - x^2) \, dx

#step 3#

We will now calculate the integral. To do this, we find the antiderivative of the function

4 - x^2

#step 4#

The antiderivative of

4 - x^2

4x - \frac{x^3}{3}

. We will evaluate this antiderivative at the bounds

x = 2

and

x = -2

and subtract the results.

#step 5#

Evaluating the antiderivative at the bounds gives us:

\left[4x - \frac{x^3}{3}\right]_{-2}^{2} = \left(4(2) - \frac{2^3}{3}\right) - \left(4(-2) - \frac{(-2)^3}{3}\right)

#step 6#

Simplifying the expression, we get:

\left(8 - \frac{8}{3}\right) - \left(-8 + \frac{8}{3}\right) = 8 - \frac{8}{3} + 8 - \frac{8}{3} = 16 - \frac{16}{3} = \frac{48}{3} - \frac{16}{3} = \frac{32}{3}

Answer

The area under the curve

y = 4 - x^2

from

x = -2

x = 2

\frac{32}{3}

square units.

Key Concept

The area under a curve between two points on the x-axis can be found by evaluating the definite integral of the function between those two points.

Explanation

The definite integral of the function

4 - x^2

from

x = -2

x = 2

represents the total area under the curve and above the x-axis between these points. By finding the antiderivative and evaluating it at the upper and lower bounds, we can calculate this area. The symmetry of the function about the y-axis simplifies the calculation, as the areas from

x = -2

x = 0

and from

x = 0

x = 2

are equal. The final result is

\frac{32}{3}

square units.