Logo

AskSia

Plus

Determine the area under the curve y=4−x2y = 4 - x^2 from x=−2x = -2 to x=2x = 2.
Apr 5, 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
Calculate the definite integral of the function y=4−x2 y = 4 - x^2 from x=−2 x = -2 to x=2 x = 2
step 2
The antiderivative of 4−x2 4 - x^2 is 4x−x33 4x - \frac{x^3}{3} plus a constant
step 3
Evaluate the antiderivative at the upper limit x=2 x = 2 to get 4(2)−233=8−83 4(2) - \frac{2^3}{3} = 8 - \frac{8}{3}
step 4
Evaluate the antiderivative at the lower limit x=−2 x = -2 to get 4(−2)−(−2)33=−8+83 4(-2) - \frac{(-2)^3}{3} = -8 + \frac{8}{3}
step 5
Subtract the value of the antiderivative at the lower limit from the value at the upper limit to find the definite integral: (8−83)−(−8+83) \left(8 - \frac{8}{3}\right) - \left(-8 + \frac{8}{3}\right)
step 6
Simplify the expression to get 16−163 16 - \frac{16}{3}
step 7
Convert the mixed number to an improper fraction: 16=483 16 = \frac{48}{3} , so 483−163=323 \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} or approximately 10.667.
Key Concept
Definite Integral as Area Under the Curve
Explanation
The definite integral of a function between two limits gives the area under the curve of the function between those two points. In this case, the area under y=4−x2 y = 4 - x^2 from x=−2 x = -2 to x=2 x = 2 is calculated using the antiderivative of the function.
What is the formula for finding the area under a curve defined by y=f(x)y = f(x) from x=ax = a to x=bx = b in terms of integration?
Solution by Steps
step 1
Identify the function to be integrated and the limits of integration
step 2
The formula for finding 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
step 3
Apply the formula to the given function and limits
Answer
The area under the curve y=f(x) y = f(x) from x=a x = a to x=b x = b is ∫abf(x) dx \int_{a}^{b} f(x) \, dx .
Key Concept
Definite Integral
Explanation
The definite integral of a function from x=a x = a to x=b x = b represents the area under the curve of that function over the interval [a, b].
Generate me a similar question
What is the process for determining the area under the curve described by y=2x+3y = 2x + 3 from x=0x = 0 to x=4x = 4 in terms of integration?
no
Solution by Steps
step 1
To find the area under the curve y=2x+3 y = 2x + 3 from x=0 x = 0 to x=4 x = 4 , we set up the definite integral
step 2
The definite integral is ∫04(2x+3) dx \int_{0}^{4} (2x + 3) \, dx
step 3
We find the antiderivative of 2x+3 2x + 3 , which is x2+3x x^2 + 3x
step 4
We evaluate the antiderivative at the upper and lower limits of integration: [(x2+3x)∣04] [(x^2 + 3x) \big|_{0}^{4}]
step 5
Plugging in the limits, we get [(42+3(4))−(02+3(0))] [(4^2 + 3(4)) - (0^2 + 3(0))]
step 6
Simplifying, we find [16+12]−[0] [16 + 12] - [0] , which equals 28 28
Answer
The area under the curve y=2x+3 y = 2x + 3 from x=0 x = 0 to x=4 x = 4 is 28 28 square units.
Key Concept
Definite Integral to Find Area
Explanation
The area under a curve y=f(x) y = f(x) from x=a x = a to x=b x = b is found by evaluating the definite integral ∫abf(x) dx \int_{a}^{b} f(x) \, dx . The result gives the total area between the curve and the x-axis within the specified interval.
© 2023 AskSia.AI all rights reserved