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Determine the area under the curve y=4−x2y = 4 - x^2 from x=−2x = -2 to x=2x = 2.
Nov 23, 2023
Determine the area under the curve y=4−x2y = 4 - x^2 from x=−2x = -2 to x=2x = 2.
Here is a Step-by-Step solution: #step 1#
Set up the integral to calculate the area under the curve from x=−2x = -2 to x=2x = 2.
#step 2#
The integral is ∫−22(4−x2) dx\int_{-2}^{2} (4 - x^2) \, dx.
#step 3#
Calculate the antiderivative of 4−x24 - x^2, which is 4x−x334x - \frac{x^3}{3}.
#step 4#
Evaluate the antiderivative from x=−2x = -2 to x=2x = 2.
#step 5#
Plug in the limits of integration: (4(2)−233)−(4(−2)−(−2)33)\left(4(2) - \frac{2^3}{3}\right) - \left(4(-2) - \frac{(-2)^3}{3}\right).
#step 6#
Simplify the expression to get the area: 8−83−(−8+83)8 - \frac{8}{3} - (-8 + \frac{8}{3}).
#step 7#
Combine terms to find the total area: 16−16316 - \frac{16}{3}.
#step 8#
Convert the mixed number to an improper fraction: 483−163\frac{48}{3} - \frac{16}{3}.
#step 9#
Subtract the fractions to get the final answer: 323\frac{32}{3}.
Answer
The area under the curve y=4−x2y = 4 - x^2 from x=−2x = -2 to x=2x = 2 is 323\frac{32}{3} square units.
Key Concept
The area under a curve in a given interval can be found by evaluating the definite integral of the function over that interval.
Explanation
The definite integral of 4−x24 - x^2 from −2-2 to 22 represents the total area under the curve and above the x-axis, minus the area above the curve and below the x-axis, between these two x-values. The calculation involves finding the antiderivative, evaluating it at the upper and lower limits, and subtracting to find the net area. The symmetry of the function about the y-axis simplifies the calculation, as the areas from −2-2 to 00 and from 00 to 22 are equal.
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