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Determine the area under the curve

y = 4 - x^2

from

x = -2

x = 2

Jan 22, 2024

Determine the area under the curve

y = 4 - x^2

from

x = -2

x = 2

Generated Graph

Solution by Steps

step 1

To find the area under the curve

y = 4 - x^2

from

x = -2

x = 2

, we need to evaluate the definite integral

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

step 2

The antiderivative of

4 - x^2

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

step 3

We evaluate the antiderivative at the upper and lower limits of the integral and subtract:

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

step 4

Plugging in the upper limit, we get

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

step 5

Plugging in the lower limit, we get

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

step 6

Subtracting the value at the lower limit from the value at the upper limit gives us

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

step 7

Simplifying the expression, we 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 Integral as Area Under the Curve

Explanation

The definite integral of a function from a to b gives the area under the curve of the function between x = a and x = b, provided the function is continuous on [a, b].