Integrals under transformations Problem set A

Given that \(\int_{0}^{7} f(x)\,dx=6.4\), find:

1. (a) \(\int_{0}^{7} 5f(x)\,dx\)
2. (b) \(\int_{0}^{7} -f(x)\,dx\)

Given that \(\int_{3}^{10} f(x)\,dx=26\):

1. (a) find \(\int_{8}^{15} g(x)\,dx\), where \(g(x)=f(x-5)\);
2. (b) find \(\int_{1}^{8} p(x)\,dx\), where \(p(x)=f(x+2)\).

Given that \(\int_{2}^{7} f(x)\,dx=11\), find:

1. (a) \(\int_{2}^{7} [f(x)+6]\,dx\)
2. (b) \(\int_{2}^{7} [f(x)-1]\,dx\)

Given that \(\int_{0}^{4} f(x)\,dx=7\):

1. (a) the graph of g is the graph of f stretched parallel to the x-axis with scale factor 3, so \(g(x)=f\left(\frac{x}{3}\right)\). Find \(\int_{0}^{12} g(x)\,dx\);
2. (b) \(h(x)=f(2x)\). Find \(\int_{0}^{2} h(x)\,dx\).

Given that \(\int_{0}^{6} f(x)\,dx=15\), the graph of g is obtained from the graph of f by a horizontal stretch with scale factor \(2\) followed by a vertical stretch with scale factor \(\frac{1}{3}\). Find \(\int_{0}^{12} g(x)\,dx\).

A tank fills at the rate of f(t) litres per minute, where t is the time in minutes after 09:00. In the first 20 minutes the tank takes in 340 litres, so \(\int_0^{20} f(t)\,dt=340\). Find the volume taken in during the first 20 minutes by each of these:

1. (a) a second tank with fill rate 3f(t);
2. (b) a third tank with fill rate f(t) + 4 (a small extra pump runs all the time);
3. (c) a fourth tank with the same pattern of filling, but starting 10 minutes later, with fill rate f(t − 10), between t = 10 and t = 30.

The power used by a factory is E(t) kilowatts, where t is the time in hours after the start of a shift. Given that \(\int_0^6 E(t)\,dt=18\), a new model for a slower shift is \(g(t)=1.5\,E\left(\frac{t}{2}\right)\). Find \(\int_0^{12} g(t)\,dt\), the energy used in kilowatt-hours over the 12 hour shift.

Given that \(\int_{0}^{4} f(x)\,dx=7.5\) and \(g(x)=2f\left(\frac{x}{3}\right)+4\), find \(\int_{0}^{12} g(x)\,dx\).

The graph of g is obtained from the graph of f by a horizontal stretch with scale factor 5 followed by a vertical stretch with scale factor 2. Given that \(\int_{0}^{15} g(x)\,dx=84\), find \(\int_{0}^{3} f(x)\,dx.\)

Given that \(\int_{1}^{5} f(x)\,dx=10\), the graph of g is obtained from the graph of f by a horizontal stretch with scale factor \(\frac{1}{2}\), then a vertical stretch with scale factor \(3\), then a translation of 4 units to the right. The interval from 1 to 5 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) \(\int_{a}^{b} [3f(x)-2]\,dx\)
2. (b) \(\int_{a+4}^{b+4} [f(x-4)+6]\,dx\)