Question textThe volume generated by rotating, about the $X$ axis, the region enclosed by $y=x^2$, $x=1,x=2$, and the $X$ axis, is Answer 1 Question 2[input] $\pi \big/$ Answer 2 Question 2[input].

Question

BlackTom AI · Accepted Answer

We are asked for the volume generated by rotating the region around the X-axis that is bounded by y = x^2, x = 1, x = 2, and the X-axis.
First, visualize the setup: the region lies between x = 1 and x = 2 and above the X-axis since y = x^2 is nonnegative in this interval.
When rotating about the X-axis, the disc (washer) method gives the volume V = π ∫_{a}^{b} [radius(x)]^2 dx, where the radius is the distance from the X-axis to the curve, i.e., radius = y = x^2.
Thus the integrand is [radius]^2 = (x^2)^2 = x^4, and the limits are from x = 1 to x = 2.
Compute the integral: ∫_{1}^{2} x^4 dx = [x^5/5]_{1}^{2} = (2^5)/5 − (1^5)/5 = 32/5 − 1/5 = 31/5.
Multiply by π for the volume: V = π × (31/5) = (31π)/5.
Now match the two blanks in the prompt: the first blank corresponds to the numerator of the fraction inside the parentheses, which is 31; the second blank corresponds to the denominator, which is 5.
So the completed form is π × (31/5) or equivalently (31π)/5, and the filled numbers are 31 and 5 respectively.
If someone mistakenly used the boundary y = x^2 as the radius without squaring or used wrong limits, they might get an incorrect coefficient, but here the proper setup and computation lead directly to the 31/5 factor times π.
In summary, the first blank should be 31, and the second blank should be 5.

ENG1090 - MUM S1 2025 Lecture quiz 11

Question textThe volume generated by rotating, about the \(X\) axis, the region enclosed by \(y=x^2\), \(x=1,x=2\), and the \(X\) axis, is Answer 1 Question 2[input] \(\pi \big/\) Answer 2 Question 2[input].

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