Exercises

1.

For each of the following functions and intervals determine the area under the curve with the int command. Also, choose one of the three rectangular approximations leftsum, middlesum or rightsum and determine the minimum number of subintervals required so that your rectangular approximation agrees with the integral to four decimal places. Don't use the same rectangular approximation method for every part; please use each method at least once. Also, plot each function to verify that it is non-negative over the interval in question.

(a)

$$f(x) = x^2 \cos(x) \mbox{ on } [-\frac{\pi}{2}, \frac{\pi}{2}]$$

(b)

$$g(x) = x^5-2x^4-x^3+16*x^2-21x+10 \mbox{ on } [0,1]$$

(c)

$$h(x) = \sqrt{4-x^2} \mbox{ on } [0,2]$$

(d)

$$r(x) = 2x (x^2+4)^{\frac{5}{2}} \mbox{ on } [0,3]$$

2.

Consider the function f(x) = -2x+8 on the interval [0,4]. Verify that the middlesum command gives the exact value for the area independently of the number of subintervals you specify. Can you explain this? Hint - look at the graph produced by the middlebox command.