Practice section 1 of 9

The diagram shows $y=x^2$ between $x=0$ and $x=1$, with four rectangles of equal width.

(i) Show that their total area is $\frac1{4^3}\sum_{r=1}^4r^2$.

(ii) If $n$ rectangles are used instead, deduce an expression for their total area.

(iii) Show that $\lim_{n\to\infty}\frac1{n^3}\sum_{r=1}^nr^2=\frac13$. What does this limit represent?

(iv) Give a reason why $\frac1{n^3}\sum_{r=1}^nr^2>\frac13$. You may use $\sum_{r=1}^nr^2=\frac n6(n+1)(2n+1)$ without proof.

Example 2 (Self-Reading). The graph $y=1/x$ on $[1,2]$ has $n$ rectangles of equal width, using right endpoints. Show that their total area is $S_n=\sum_{r=1}^n\frac1{n+r}$ and find the exact value of $\lim_{n\to\infty}S_n$. Teacher check: construct a similar diagram using left endpoints.

Example 3 (Self-Reading). Without the use of a calculator, evaluate (a) $\int_0^2 3(1+x)^2\,dx$, (b) $\int_0^{\pi/2}\cos x\,dx$, (c) $\int_1^2e^{1+x}\,dx$.

Without the use of a calculator, evaluate (a) $\int_0^2 5(2x-1)^4\,dx$, (b) $\int_{-1}^0\frac1{\sqrt{3x+14}}\,dx$.

Evaluate $\int_4^8\frac{x+1}{x\sqrt{x-2}}\,dx$ exactly, using the substitution $u=\sqrt{x-2}$.

Evaluate $\int_1^2\ln x\,dx$ exactly.