Integrals under transformations Problem set B

Given that \(\int_{1}^{6} f(x)\,dx=2.8\), find:

1. (a) \(\int_{1}^{6} 6f(x)\,dx\)
2. (b) \(\int_{1}^{6} -f(x)\,dx\)

Given that \(\int_{1}^{6} f(x)\,dx=14.5\):

1. (a) find \(\int_{5}^{10} g(x)\,dx\), where \(g(x)=f(x-4)\);
2. (b) find \(\int_{-2}^{3} p(x)\,dx\), where \(p(x)=f(x+3)\).

Given that \(\int_{1}^{5} f(x)\,dx=8.5\), find:

1. (a) \(\int_{1}^{5} [f(x)+4]\,dx\)
2. (b) \(\int_{1}^{5} [f(x)-3]\,dx\)

Given that \(\int_{0}^{5} f(x)\,dx=6\):

1. (a) the graph of g is the graph of f stretched parallel to the x-axis with scale factor 4, so \(g(x)=f\left(\frac{x}{4}\right)\). Find \(\int_{0}^{20} g(x)\,dx\);
2. (b) \(h(x)=f(5x)\). Find \(\int_{0}^{1} h(x)\,dx\).

Given that \(\int_{0}^{9} f(x)\,dx=12\), the graph of g is obtained from the graph of f by a horizontal stretch with scale factor \(\frac{1}{3}\) followed by a vertical stretch with scale factor \(5\). Find \(\int_{0}^{3} g(x)\,dx\).

Customers enter a shop at the rate of f(t) per hour, where t is the time in hours after the shop opens. In the first 6 hours 210 customers come in, so \(\int_0^6 f(t)\,dt=210\). Find the number of customers in the first 6 hours for each of these:

1. (a) a bigger shop with customer rate 2.5f(t);
2. (b) a shop with an advert that brings an extra 7 customers every hour: rate f(t) + 7;
3. (c) a branch that opens 3 hours later with the same pattern, rate f(t − 3), between t = 3 and t = 9.

The rate of rainfall during a storm is R(t) millimetres per hour, where t is the time in hours after it starts. Given that \(\int_0^4 R(t)\,dt=1.6\), a model for a longer, heavier storm is \(g(t)=2.5\,R\left(\frac{t}{3}\right)\). Find \(\int_0^{12} g(t)\,dt\), the total rainfall in millimetres over the 12 hours.

Given that \(\int_{0}^{5} f(x)\,dx=4.2\) and \(g(x)=3f\left(\frac{x}{2}\right)-1\), find \(\int_{0}^{10} g(x)\,dx\).

The graph of g is obtained from the graph of f by a horizontal stretch with scale factor 3 followed by a vertical stretch with scale factor 4. Given that \(\int_{0}^{12} g(x)\,dx=150\), find \(\int_{0}^{4} f(x)\,dx.\)

Given that \(\int_{3}^{9} f(x)\,dx=12\), the graph of g is obtained from the graph of f by a horizontal stretch with scale factor \(\frac{1}{3}\), then a vertical stretch with scale factor \(2\), then a translation of 5 units to the left. The interval from 3 to 9 on the graph of f becomes the interval from p to q on the graph of g.

1. (a) Find p and q.
2. (b) Find \(\int_{p}^{q} g(x)\,dx\).

Given that \(\int_a^b f(x)\,dx=X\), find, in terms of a, b and X:

1. (a) \(\int_{a}^{b} [4f(x)+5]\,dx\)
2. (b) \(\int_{a-3}^{b-3} [f(x+3)-2]\,dx\)