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Determine the area under the curve y=4−x2y = 4 - x^2 from x=−2x = -2 to x=2x = 2.
Feb 27, 2024
Determine the area under the curve y=4−x2y = 4 - x^2 from x=−2x = -2 to x=2x = 2.
Generated Graph
Solution by Steps
step 1
Set up the definite integral for the area under the curve y=4−x2 y = 4 - x^2 from x=−2 x = -2 to x=2 x = 2
step 2
Calculate the antiderivative of 4−x2 4 - x^2 . The antiderivative is 4x−x33+C 4x - \frac{x^3}{3} + C , where C C is the constant of integration
step 3
Evaluate the antiderivative from x=−2 x = -2 to x=2 x = 2 . This gives us [4(2)−233]−[4(−2)−(−2)33] \left[ 4(2) - \frac{2^3}{3} \right] - \left[ 4(-2) - \frac{(-2)^3}{3} \right]
step 4
Simplify the expression to find the area. [8−83]−[−8+83]=[8−83+8−83]=[16−163] \left[ 8 - \frac{8}{3} \right] - \left[ -8 + \frac{8}{3} \right] = \left[ 8 - \frac{8}{3} + 8 - \frac{8}{3} \right] = \left[ 16 - \frac{16}{3} \right]
step 5
Convert the mixed number to an improper fraction and simplify to get the final answer. 16−163=483−163=323 16 - \frac{16}{3} = \frac{48}{3} - \frac{16}{3} = \frac{32}{3}
Answer
323 \frac{32}{3} or approximately 10.667
Key Concept
Definite Integral to Find Area
Explanation
The area under the curve y=f(x) y = f(x) from x=a x = a to x=b x = b is given by the definite integral ∫abf(x) dx \int_{a}^{b} f(x) \, dx . In this case, the function is 4−x2 4 - x^2 , and the limits of integration are -2 and 2.
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