Find the area under the curve y = x^4 from x = 0 to x = 1.

₹0.0

Question: Find the area under the curve y = x^4 from x = 0 to x = 1.

Options:

Correct Answer: 1/5

Solution:

The area under the curve y = x^4 from 0 to 1 is given by ∫(from 0 to 1) x^4 dx = [x^5/5] from 0 to 1 = 1/5.

Find the area under the curve y = x^4 from x = 0 to x = 1.

Find the area under the curve y = x^4 from x = 0 to x = 1.

Step 1: Understand that we want to find the area under the curve of the function y = x^4 between x = 0 and x = 1.
Step 2: To find the area under the curve, we need to calculate the definite integral of the function from 0 to 1.
Step 3: Write the integral as ∫(from 0 to 1) x^4 dx.
Step 4: Find the antiderivative of x^4. The antiderivative is (x^5)/5.
Step 5: Now, we need to evaluate this antiderivative from 0 to 1. This means we will calculate (1^5)/5 - (0^5)/5.
Step 6: Calculate (1^5)/5, which is 1/5, and (0^5)/5, which is 0.
Step 7: Subtract the two results: 1/5 - 0 = 1/5.
Step 8: Therefore, the area under the curve y = x^4 from x = 0 to x = 1 is 1/5.

Definite Integral – The process of calculating the area under a curve between two points using integration.
Polynomial Functions – Understanding the behavior and properties of polynomial functions, specifically x^4 in this case.