FAD1014: MATHEMATICS II — Tutorial 6

Centre for Foundation Studies in Science, University Malaya
Session 2025/2026

Question 1: Volume of Revolution About X-Axis

(a) Sketch and shade the following regions R.

(b) Find the volume of the solid of revolution formed when the area R is rotated about the x-axis.

(i) R is bounded by $y = x$, $y = 0$, $x = 1$ and $x = 2$

Answer: $\frac{3\pi}{2}$ unit³

(ii) R is bounded by $y = -x^2$, $y = 0$ on the interval $[2, 5]$

Answer: $\frac{3093\pi}{5}$ unit³

(iii) R is bounded by $y = \sin^2 x$, $y = 0$ and $x \in [0, \frac{\pi}{2}]$

Answer: $\frac{\pi^2}{4}$ unit³

Question 2: Region Bounded by Curves

The region R is bounded by the curves $y = x^2$ $(x \geq 0)$, $y = 9x^2$ $(x \geq 0)$ and the line $y = 1$.

(a) Sketch and shade the region R.

(b) Calculate the area R.

(c) Find the volume of the solid obtained when the region R is rotated about the y-axis.

Question 3: Intersection of Parabolas

The area A is the region bounded by the curves $x^2 = 12y$ and $y^2 = 12x$.

(a) Calculate the area A.

(b) Find the volume of the solid formed when the region A is rotated about the x-axis.

Question 4: Region Enclosed by Curve and Line

The area R is the region enclosed by the curve $y = x^2$ and the line $y = x$.

(a) Show that the area obtained is $\frac{1}{6}$ unit².

(b) Find the volume of the solid formed if the area R is rotated about the:

  • (i) x-axis
  • (ii) the line $x = 2$

Question 5: Volume by Revolution

Calculate the volume of the solid formed when the region bounded by $y = x^2 + 2$, $2y - x = 2$, $x = 0$ and $x = 2$ is revolved around the x-axis.

Question 6: Volume About Vertical Lines

The region R is bounded by the curves $y = x^2$ $(x \geq 0)$, $y = 9x^2$ $(x \geq 0)$ and the line $y = 1$. Find the volume of the solid obtained when the region R is rotated about the line:

(a) $x = 1$

(b) $x = 2$

Question 7: Region Bounded by Curve, Line and Axis

The region R is bounded by the curve $y = x^2$ $(x \geq 0)$, the line $y = 8 - 2x$ and $y = 0$.

(a) Find the intersection point of the curve and the line $y = 8 - 2x$.

(b) Calculate the area R and the volume of the solid obtained when the region R is rotated about the x-axis.

Related Concepts

  • Integration Techniques
  • Volumes of Revolution
  • Disk Method
  • Washer Method
  • Shell Method
  • Definite Integrals
  • Solid of Revolution

Related Lectures

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