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

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

Options:

Correct Answer: 16

Solution:

The area is given by the integral from 0 to 2 of x^4 dx. This evaluates to [x^5/5] from 0 to 2 = (32/5) = 16.

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

Find the area under the curve y = x^4 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^4.
Step 2: Determine the limits of integration. We want to find the area from x = 0 to x = 2.
Step 3: Set up the integral to find the area. This is written as the integral from 0 to 2 of x^4 dx.
Step 4: Calculate the integral. The integral of x^4 is (x^5)/5.
Step 5: Evaluate the integral at the upper limit (x = 2) and the lower limit (x = 0).
Step 6: For the upper limit, substitute x = 2 into (x^5)/5: (2^5)/5 = 32/5.
Step 7: For the lower limit, substitute x = 0 into (x^5)/5: (0^5)/5 = 0.
Step 8: Subtract the lower limit result from the upper limit result: (32/5) - 0 = 32/5.
Step 9: The final answer is 32/5, which can also be expressed as 6.4.

Definite Integral – The process of calculating the area under a curve by evaluating the integral of a function over a specified interval.
Polynomial Functions – Understanding the properties and behavior of polynomial functions, specifically how to integrate them.