A. Find the integral of the following function.
Integral of (x√(x+1)) dx.
B. Set up and evaluate the integral of (2√x) for the area of the surface generated by revolving the curve about the x-axis from 4 to 9.
For part B of our question , the surface of revolution is is
integral of 2 pi y * dx
= 4 pi x^(1/2) dx
for x from 4 to 9.
The indefinite integral is
4 pi x^(3/2)/(3/2)
For the definite integral, subtract the value at x = 4 from the value at x = 9.
Try making the subsitution u = x + 1
x = u -1
dx = du
The integral becomes
Integral of (u^3/2) - u^(1/2) du
= (2/5) [u^(5/2) - (2/3) u^(3/2)] du
= (2/5) (x+1)^(5/2) - (2/3)(x+1)^(3/2)]
For the part B of you question, integrate
2 sq
To find the integral of the function (x√(x+1)) dx, we can use integration techniques. Let's break it down step by step.
Step 1: Rewrite the function using a power rule. Notice that the function has a polynomial multiplied by the square root of a binomial. We can rewrite it as:
(x(x+1)^(1/2)) dx
Step 2: Simplify the expression. Multiply x by the square root term:
x^(3/2)(x+1)^(1/2) dx
Step 3: Apply the power rule for integration. The power rule states that the integral of x^n dx, where n is any real number except -1, is (x^(n+1))/(n+1).
Using the power rule, we can find the integral of x^(3/2)(x+1)^(1/2) dx:
= (2/5)(x+1)^(5/2) + C
So, the integral of (x√(x+1)) dx is (2/5)(x+1)^(5/2) + C, where C is the constant of integration.