Which of the following integrals correctly computes the volume formed when the region bounded by the curves x^2 + y^2 = 25, x = 4 and y = 0 is rotated around the y-axis?
A. pi ∫ upper bound of 3 and lower bound of 0 ( sqrt(25 - y^2) -4)^2 dy
B. pi ∫ upper bound of 3 and lower bound of 0 ( 4^2 - ( sqrt(25-y^2) )^2 ) dy
C. pi ∫ upper bound of 5 and lower bound of 4 ( sqrt(25-x^2) )^2 dy
D. pi ∫ upper bound of 3 and lower bound of 0 ( (sqrt(25-y^2)^2 ) - 4^2 ) dy
I'm struggling on this, please helpp!
Thank you so so much in advance!
The region being rotated is the triangular slice with vertices at (4,0), (4,3), (5,0). Recall that for washers, the volume is
v = ∫ π(R^2-r^2) dy
Here, R is the circle, and r is the line x=4. So, choice (D)
choice d is right
To determine the correct integral that computes the volume formed when the region bounded by the curves x^2 + y^2 = 25, x = 4, and y = 0 is rotated around the y-axis, we need to find the correct setup for the integral.
First, let's visualize the region and the rotation around the y-axis. The curve x^2 + y^2 = 25 represents a circle with a radius of 5 centered at the origin (0,0). The line x = 4 is a vertical line to the right of the circle, and the line y = 0 is the x-axis.
The region that we are interested in is the area between the curve x^2 + y^2 = 25 and the line x = 4, bounded by the x-axis and y-axis. We need to rotate this region around the y-axis to find the volume.
To set up the integral correctly, we are integrating with respect to y. The correct setup involves expressing the radius of each cross section (perpendicular to the y-axis) of the region as a function of y.
Let's go through each option:
A. pi ∫ upper bound of 3 and lower bound of 0 ( sqrt(25 - y^2) - 4)^2 dy.
This option doesn't correctly represent the radius of each cross section since it subtracts 4 from the radius when squaring it.
B. pi ∫ upper bound of 3 and lower bound of 0 (4^2 - (sqrt(25-y^2))^2) dy.
This option correctly represents the radius of each cross section. It is the difference between the radius of the circle (5) and the y-coordinate of the curve at that point (sqrt(25-y^2)).
C. pi ∫ upper bound of 5 and lower bound of 4 (sqrt(25-x^2))^2 dy.
This option incorrectly sets up the integral with respect to y but expresses the radius of each cross section with respect to x instead.
D. pi ∫ upper bound of 3 and lower bound of 0 ((sqrt(25-y^2))^2 - 4^2) dy.
This option incorrectly represents the radius of each cross section since it subtracts 4^2 from the squared radius.
Therefore, the correct integral that computes the volume formed when the region is rotated around the y-axis is option B:
pi ∫ upper bound of 3 and lower bound of 0 (4^2 - (sqrt(25-y^2))^2) dy.
To find the volume formed when the given region is rotated around the y-axis, we can use the method of cylindrical shells. The formula for the volume using cylindrical shells is:
V = ∫[a, b] 2πx * h(x) dx,
where [a, b] is the interval along the x-axis that includes the region, x represents the values along the x-axis, and h(x) represents the height of each cylindrical shell.
Let's go through each option to see which one correctly represents the volume integral for the given problem:
Option A: pi ∫[3, 0] (sqrt(25 - y^2) - 4)^2 dy.
This integral represents the volume using y as the variable of integration. However, we need to use x as the variable since we are integrating along the x-axis. So, this option is not correct.
Option B: pi ∫[3, 0] (4^2 - (sqrt(25-y^2))^2) dy.
This expression represents the volume integral as required. It correctly uses y as the variable of integration and subtracts the square of the radius from the square of the constant value of x (x = 4) within the integral. This option seems to be the correct answer.
Option C: pi ∫[5, 4] (sqrt(25-x^2))^2 dy.
This integral represents integrating along the y-axis instead of the x-axis, so it is not correct for our problem.
Option D: pi ∫[3, 0] ((sqrt(25-y^2))^2 - 4^2) dy.
This expression correctly uses y as the variable of integration but evaluates the incorrect function within the integral. The expression (sqrt(25-y^2))^2 - 4^2 does not represent the radius of the cylindrical shells correctly, so this option is not correct.
Therefore, the correct option is B, which is:
pi ∫[3, 0] (4^2 - (sqrt(25-y^2))^2) dy.
This represents the volume formed when the given region is rotated around the y-axis.