Questions 2 (Hints and solutions start on page .)

Q 2.1   Work out the area between the following graphs and the x-axis on the given ranges:

y = x⁴     on     [1, 2]

y = e^x     on     [- 1, 1]

$${\frac{\pi}{2}} {\frac{\pi}{2}}$$ y = cos x

y = 2x² + x     on     [0, 1]

y = x² - x     on     [0, 2]

y = x² - 3x + 2     on     [0, 3]

y = x³     on     [- 2, 1]

$$\infty$$ y = e^-x

Q 2.2   Find the area enclosed between the graphs of y = x² and y = x + 2 by first sketching the graphs to see what is going on and then working out the points at which the two graphs meet before doing an integration.

Q 2.3   Find where the graphs of y = x(1 - x) and y = x² cross and then find the area enclosed between the two graphs.

Q 2.4   Find the area enclosed by the graphs y = e^x, y = x² and the lines x = 0 and x = 1.

Q 2.5   Consider the area enclosed between the graph of y = 1 - x² and the x-axis. Which line parallel to the x-axis divides this area into two equal parts?

*Q 2.6  

Consider the function b(x) defined by

Remembering the basic fact that the definite integral gives the area under a graph (taking due account of signs), draw the graph of the function b₁(x) defined by

b₁(x) = $$\int_{0}^{x}$$b(t) dt

for all values of x.

If you managed that you can now go on to sketch the graph of

b₂(x) = $$\int_{0}^{x}$$b₁(t) dt

*Q 2.7   Suppose that f (x) and g(x) are two functions and consider the integral

I(t) = $$\int_{a}^{b}$$(f (x) + tg(x))² dx.

whose value depends on t. Because of the square, this integral cannot be negative: I(t)$\ge$ 0. Now expand out the square and show that you get I(t) = At² + 2Bt + C where A, B, C are constants. Use the fact that I(t) cannot be negative, together with what you know about quadratics, to show that you must have B²$\le$AC and that this becomes

$$\left(\vphantom{\int_a^b f(x)g(x) dx}\right.$$$$\int_{a}^{b}$$f (x)g(x) dx$$\left.\vphantom{\int_a^b f(x)g(x) dx}\right)^{2}_{}$$$$\le$$$$\int_{a}^{b}$$f (x)² dx ^. $$\int_{a}^{b}$$g(x)² dx    (Cauchy-Schwartz Inequality)     

This is a very useful inequality in more advanced work.

*Q 2.8   This exercise relates to the method that Archimedes used to find the area of the region between a parabola (let's say y = x²) and a chord of the parabola. He did this nearly 2000 years before the calculus was officially invented.

Let P(a, a²) and Q(b, b²) be two points on the graph y = x² with b > a. Let R be the point on the graph where the slope of the graph is equal to the slope of the chord PQ. Show that R is the point on the graph with x-coordinate (a + b)/2. Why is PQR the biggest triangle with base PQ and with third vertex on the graph between P and Q? The area of the triangle PQR is (b - a)³/8 (as you may be able to check). Now find the area of the `parabola segment' bounded by the parabola and the chord and show that it comes out as 4/3 times the area of the triangle.

*Q 2.9  

This question is about estimating the value of $\pi$. Draw the graph of y = sin x for 0$\le$x$\le$$\pi$/2. Now work out the area between this graph and the x-axis. By looking at your picture can you show that 2 < $\pi$ < 4. Not very precise, but it's a start. By looking at the area of the graph between x = 0 and x = $\pi$/4 can you show that $\pi$ < 8($\sqrt{2}$ - 1) and, by using the fact that sin x$\le$x on this range, that $\pi^{2}_{}$ > 32(1 - 1/$\sqrt{2}$)?

*Q 2.10  

The Harmonic Numbers H₁, H₂, H₃,... are defined by

H_n = 1 + $${\textstyle\frac{1}{2}}$$ + $${\textstyle\frac{1}{3}}$$ + $${\textstyle\frac{1}{4}}$$ + $${\textstyle\frac{1}{5}}$$ + ^... + $${\frac{1}{n}}$$

So, for example, H₁ = 1, H₂ = 3/2, H₃ = 11/6, etc.

Study the Fig. 2.5 which shows the graph of y = 1/x:

Figure 2.5: Estimating an area.

Remember that the area under the graph between x = 1 and x = n is given by the integral $\int_{1}^{n}$dx/x = ln(n). Add up the areas of the rectangles and show that

$${\textstyle\frac{1}{2}} {\textstyle\frac{1}{3}} {\frac{1}{n-1}}$$ UPPER

$${\textstyle\frac{1}{2}} {\textstyle\frac{1}{3}} {\textstyle\frac{1}{4}} {\frac{1}{n}}$$ LOWER

Deduce from this that for n > 1,

$$\boxed{\ln(n)+\frac{1}{n} < H_n < \ln(n)+1}$$

Notice that this implies that H_n$\to$$\infty$ as n$\to$$\infty$, rather surprisingly. Roughly what value of n do you need to get H_n = 10? What about H_n = 20 or H_n = 100?

The above calculation narrowed the value of H_n down to a range of length about 1. We can do a lot better than this. The main reason for the roughness in the approximation was that the first few rectangles fitted the graph very badly. So suppose we start the exercise further along. By applying the above process to the graph on the range N$\le$x$\le$n show that

ln(n) + $${\frac{1}{n}}$$ + A_N - $${\frac{1}{N}}$$ < H_n < ln(n) + A_N

where A_N = H_N - ln(N). Suppose I tell you that H₁₀₀₀ = 7.4854708606  to 10 decimal places. What does the above inequality become and how accurately can you tell me the value of H_1000000 ?