Can someone check my answers:
1) Use geometry to evaluate 6 int 2 (x) dx where f(x) = { |x|, -2 <= x <= 2}
{2, 2 < x <= 4}
{-x+4 4 < x <= 6}
for my answer I got 0.512
2) The base of a solid is bounded by y = √x+2, the x-axis and the line x = 1. The cross sections, taken perpendicular to the x-axis, are squares. Find the volume of the solid.
a) 0.5
b) 3.464 <-- my answer
c) 4.5
d) None of the above
Thank you!
1) To evaluate the integral using geometry, we need to find the area under the given function f(x) over the specified intervals. Let's break it down step by step:
Step 1: Evaluate the area under the function f(x) = |x| from -2 to 2:
This is a triangle with a base of 4 and a height of 2. The area of a triangle is given by (1/2) * base * height, so the area under this portion is (1/2) * 4 * 2 = 4.
Step 2: Evaluate the area under the function f(x) = 2 from 2 to 4:
This is a rectangle with a base of 2 and a height of 2. The area of a rectangle is given by base * height, so the area under this portion is 2 * 2 = 4.
Step 3: Evaluate the area under the function f(x) = -x+4 from 4 to 6:
This is also a triangle with a base of 2 and a height of 2. The area under this portion is (1/2) * 2 * 2 = 2.
Step 4: Add up the areas from all three parts to get the total area:
Total area = 4 + 4 + 2 = 10.
Therefore, the correct answer for the integral is 10, not 0.512. Please re-check your calculations.
2) To find the volume of the solid, we need to integrate the area of each cross section perpendicular to the x-axis.
Step 1: The cross sections are squares with side length given by the function f(x) = √(x+2).
Step 2: To find the limits of integration, we set √(x+2) = 0 (cross section intersects the x-axis) and √(x+2) = 1 (cross section intersects x = 1).
Setting √(x+2) = 0, we find that x = -2.
Setting √(x+2) = 1, we find that x = -1.
So, the limits of integration are from x = -2 to x = -1.
Step 3: Integrate the area of each cross section from -2 to -1.
The area of a square with side length √(x+2) is (side length)^2 = (√(x+2))^2 = x + 2.
Integrating x + 2 from -2 to -1, we get [(x^2)/2 + 2x] from -2 to -1.
Plugging in the limits, we have [((-1)^2)/2 + 2(-1)] - [((-2)^2)/2 + 2(-2)].
Simplifying this, we get (1/2 - 2) - (2/2 + 4) = (-3/2) - (5/2) = -8/2 = -4.
Therefore, the volume of the solid is -4 (since it is below the x-axis in this case), not 3.464. Please re-check your calculations.