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Determine the area under the curve

y = 4 - x^2

from

x = -2

x = 2

May 26, 2024

Determine the area under the curve

y = 4 - x^2

from

x = -2

x = 2

Generated Graph

Solution by Steps

step 1

Calculate the definite integral of the function

y = 4 - x^2

from

x = -2

x = 2

step 2

The antiderivative of

4 - x^2

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

plus a constant

step 3

Evaluate the antiderivative at the upper limit

x = 2

to get

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

step 4

Evaluate the antiderivative at the lower limit

x = -2

to get

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

step 5

Subtract the value of the antiderivative at the lower limit from the value at the upper limit to find the definite integral:

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

step 6

Simplify the expression to get

16 - \frac{16}{3} = \frac{48}{3} - \frac{16}{3} = \frac{32}{3}

Answer

\frac{32}{3}

or approximately 10.667

Key Concept

Definite Integration to Find Area

Explanation

The area under the curve

y = 4 - x^2

from

x = -2

x = 2

is found by evaluating the definite integral of the function over the given interval. The result is the total area between the curve and the x-axis within the limits of integration.

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