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Determine the area under the curve

y = 4 - x^2

from

x = -2

x = 2

Nov 22, 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 evaluate the integral. First, we find the antiderivative of the function

4 - x^2

#step 4#

The antiderivative of

4 - x^2

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

#step 5#

We now apply the Fundamental Theorem of Calculus to evaluate the definite integral from

x = -2

x = 2

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

#step 6#

Plugging in the upper bound, we get:

4(2) - \frac{2^3}{3} = 8 - \frac{8}{3}

#step 7#

Plugging in the lower bound, we get:

4(-2) - \frac{(-2)^3}{3} = -8 + \frac{8}{3}

#step 8#

Subtracting the value at the lower bound from the value at the upper bound gives us:

\left(8 - \frac{8}{3}\right) - \left(-8 + \frac{8}{3}\right)

#step 9#

Simplifying the expression, we find the area under the curve:

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 representing the curve between those two points.

Explanation

The solution involved setting up the definite integral for the function

4 - x^2

between the limits of integration

-2

and

2

, finding the antiderivative, and then evaluating the definite integral to find the area under the curve. The area is represented by the difference in the antiderivative evaluated at the upper and lower bounds of integration.