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filna555's Question
Math
Posted about 1 month ago
002848
Determine the area under the curve y=4x2y = 4 - x^2 from x=2x = -2 to x=2x = 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=4x2y = 4 - x^2 from x=2x = -2 to x=2x = 2, we need to evaluate the definite integral: 22(4x2)dx\int_{-2}^{2} (4 - x^2) \, dx
step 2
Calculating the integral, we have: (4x2)dx=4xx33+C\int (4 - x^2) \, dx = 4x - \frac{x^3}{3} + C
step 3
Now, we evaluate the definite integral from 2-2 to 22: [4(2)(2)33][4(2)(2)33][4(2) - \frac{(2)^3}{3}] - [4(-2) - \frac{(-2)^3}{3}]
step 4
This simplifies to: (883)(8+83)=(883)+(883)=16163=483163=323(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=2x = -2 to x=2x = 2 is 323\frac{32}{3} or approximately 10.66710.667.
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=4x2y = 4 - x^2 from $x = -

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