Calculate the area under the curve y = x^3 from x = 0 to x = 2.

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Question: Calculate the area under the curve y = x^3 from x = 0 to x = 2.

Options:

Correct Answer: 8

Solution:

The area under the curve is given by ∫(from 0 to 2) x^3 dx = [x^4/4] from 0 to 2 = (16/4) - (0) = 4.

Calculate the area under the curve y = x^3 from x = 0 to x = 2.

Calculate the area under the curve y = x^3 from x = 0 to x = 2.

Step 1: Identify the function you want to find the area under. In this case, the function is y = x^3.
Step 2: Set up the integral to calculate the area under the curve from x = 0 to x = 2. This is written as ∫(from 0 to 2) x^3 dx.
Step 3: Find the antiderivative of the function x^3. The antiderivative is x^4 / 4.
Step 4: Evaluate the antiderivative at the upper limit (x = 2) and the lower limit (x = 0).
Step 5: Calculate the value at the upper limit: (2^4) / 4 = 16 / 4 = 4.
Step 6: Calculate the value at the lower limit: (0^4) / 4 = 0 / 4 = 0.
Step 7: Subtract the lower limit value from the upper limit value: 4 - 0 = 4.
Step 8: The area under the curve from x = 0 to x = 2 is 4.

Definite Integral – The process of calculating the area under a curve by evaluating the integral of a function over a specified interval.
Power Rule for Integration – A method used to integrate polynomial functions, where the integral of x^n is (x^(n+1))/(n+1) + C.