3. Find the volume of the solid with cross-sectional area A(x) = 10 e^0.01x, 0 ≤ x ≤ 10.
e^x is non-negative, so the volume is the definite integral from 0 to 10 of
I=∫A(x)dx
=0.01*10e^0.01x
=0.1*e^0.01x
evaluated from 0 to 10
To find the volume of the solid, we need to integrate the cross-sectional area function A(x) over the given interval [0, 10]. Here's how you can do it step-by-step:
Step 1: Express the cross-sectional area function in terms of x.
The given cross-sectional area function is A(x) = 10 e^(0.01x).
Step 2: Set up the integral.
The volume of the solid is calculated by integrating the area function with respect to x over the interval [0, 10]. So, the integral to find the volume (V) can be set up as follows:
V = ∫[0,10] A(x) dx
Step 3: Evaluate the integral.
Now, we need to calculate the integral of A(x) over the interval [0, 10]. Let's perform the antiderivative of A(x) and use the exponential rule of integration:
V = 10 ∫[0,10] e^(0.01x) dx
Step 4: Evaluate the integral limits.
Substitute the upper limit (10) into the antiderivative expression and subtract the result obtained by substituting the lower limit (0):
V = 10 [e^(0.01 * 10) - e^(0.01 * 0)]
Simplify the expression:
V = 10 [e^0.1 - e^0]
Since e^0 equals 1, we can simplify it further:
V = 10 [e^0.1 - 1]
Now, calculate the value of e^0.1 using a calculator or an approximation:
V ≈ 10 [1.105170918 - 1]
V ≈ 10 [0.105170918]
V ≈ 1.05170918 (approximately)
Therefore, the volume of the solid with the given cross-sectional area function is approximately 1.05170918 cubic units.