Calculate the area under the curve y = x^4 from x = 0 to x = 2.

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Question: Calculate the area under the curve y = x^4 from x = 0 to x = 2.

Options:

Correct Answer: 8

Solution:

The area under the curve y = x^4 from x = 0 to x = 2 is given by ∫(from 0 to 2) x^4 dx = [x^5/5] from 0 to 2 = (32/5) - 0 = 32/5.

Calculate the area under the curve y = x^4 from x = 0 to x = 2.

Calculate the area under the curve y = x^4 from x = 0 to x = 2.

Correct Answer: 32/5

Step 1: Identify the function you want to find the area under. In this case, the function is y = x^4.
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^4 dx.
Step 3: Find the antiderivative of the function x^4. The antiderivative is x^5 / 5.
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: (2^5) / 5 = 32 / 5.
Step 6: Calculate the value at the lower limit: (0^5) / 5 = 0.
Step 7: Subtract the lower limit value from the upper limit value: (32/5) - 0 = 32/5.
Step 8: The final result is the area under the curve from x = 0 to x = 2, which is 32/5.

Definite Integral – The process of calculating the area under a curve using integration over a specified interval.
Polynomial Functions – Understanding the behavior and properties of polynomial functions, specifically y = x^4 in this case.
Fundamental Theorem of Calculus – Applying the fundamental theorem to evaluate the definite integral by finding the antiderivative.