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Determine the area under the curve y=4−x2y = 4 - x^2 from x=−2x = -2 to x=2x = 2.
Jan 2, 2024
Determine the area under the curve y=4−x2y = 4 - x^2 from x=−2x = -2 to x=2x = 2.
Solution by Steps
step 1
To find the area under the curve y=4−x2 y = 4 - x^2 from x=−2 x = -2 to x=2 x = 2 , we need to integrate the function with respect to x x over the given interval
step 2
The antiderivative of 4−x2 4 - x^2 is 4x−x33 4x - \frac{x^3}{3} , using the power rule for integration
step 3
We evaluate the antiderivative from x=−2 x = -2 to x=2 x = 2 . This is done by calculating [4x−x33]−22 \left[4x - \frac{x^3}{3}\right]_{-2}^{2}
step 4
Substituting the upper limit of the integral, we get 4(2)−233=8−83 4(2) - \frac{2^3}{3} = 8 - \frac{8}{3}
step 5
Substituting the lower limit of the integral, we get 4(−2)−(−2)33=−8+83 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 (8−83)−(−8+83) \left(8 - \frac{8}{3}\right) - \left(-8 + \frac{8}{3}\right)
step 7
Simplifying the expression, we get 8−83+8−83=16−163=483−163=323 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 y=4−x2 y = 4 - x^2 from x=−2 x = -2 to x=2 x = 2 is 323 \frac{32}{3} square units.
Key Concept
Integration to find the area under a curve
Explanation
The area under the curve y=4−x2 y = 4 - x^2 from x=−2 x = -2 to x=2 x = 2 is found by integrating the function and evaluating the definite integral 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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