Answer all question

Assignment 3 Due : 4pm Tuesday November 4 OMAS120 Applied Mathematics Study Period 3, 2014

Mathematics and Statistics

1. Find the following integrals. Make sure you use calculus and show all your working.

(i)

∫ 2x √

4x2 + 3 dx

(ii)

∫ √ t

(t3/2 − 4)4 dt

(iii)

∫ 2 0

z−1/2 √

4 + 2 √ z dz

2. Using calculus and showing all your working find the area bounded by the curves f(x) = x3 −x2 + x and g(x) = 2x2 −x.

3. Consider the integral

∫ 2 0

√ x + 4 dx. Using the trapezoidal method with n = 4

and n = 8, estimate the integral numerically. Calculate the integral exactly (using calculus and showing your working) and compare this with your numerical results.

4. Find the following integrals. Show all working.

(i)

∫ 1 0

(x2 − 2x)e2x dx

(ii)

∫ (2x− 2) ln(x) dx

5. Find the average value of the function f(x) = √

4x− 2 over x ∈ [1, 4].

6. A ‘frustum’ is what remains of a cone when the top is cut off parallel to the base. Suppose a frustum has base radius R, height h and top radius r. By considering a suitable function and using volumes of revolution, derive the formula for the volume

of the frustum V = 1

3 πh(R2 + Rr + r2).

Notes:

• 10% of the marks for this assignment are reserved for presentation.

• There are penalties for late assignments. You must contact your tutor before the due date if you have difficulties making the deadline.

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