002848

`Determine the area under the curve $y = 4 - x^2$ from $x = -2$ to $x = 2$.`

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Answer from Sia

Posted about 1 month ago

Solution by Steps

step 1

To find the area under the curve $y = 4 - x^2$ from $x = -2$ to $x = 2$, we need to evaluate the definite integral: $\int_{-2}^{2} (4 - x^2) \, dx$

step 2

Calculating the integral, we have: $\int (4 - x^2) \, dx = 4x - \frac{x^3}{3} + C$

step 3

Now, we evaluate the definite integral from $-2$ to $2$: $[4(2) - \frac{(2)^3}{3}] - [4(-2) - \frac{(-2)^3}{3}]$

step 4

This simplifies to: $(8 - \frac{8}{3}) - (-8 + \frac{8}{3}) = (8 - \frac{8}{3}) + (8 - \frac{8}{3}) = 16 - \frac{16}{3} = \frac{48}{3} - \frac{16}{3} = \frac{32}{3}$

Answer

The area under the curve from $x = -2$ to $x = 2$ is $\frac{32}{3}$ or approximately $10.667$.

Key Concept

The area under a curve can be found using definite integrals.

Explanation

The definite integral calculates the net area between the curve and the x-axis over a specified interval. In this case, it gives the total area under the curve $y = 4 - x^2$ from $x = -

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