Let and be continuous functions on the interval with for every in that interval

MATHEMATICS · Multi-part · A Level · Mathematics · Integration · Integration

Problem

Let

h

and

k

be continuous functions on the interval

1\le x\le 5

with

h(x)\ge k(x)

for every

x

in that interval. The region between the graphs of

y=h(x)

and

y=k(x)

from

x=1

x=5

has area

10

square units. Define new functions

p

and

q

p(x)=3h(x)

and

q(x)=3k(x)-4

for

1\le x\le 5

. (a) Explain why

\displaystyle\int_{1}^{5}\bigl(h(x)-k(x)\bigr)\,dx=10

. State any properties of definite integrals you use. (b) Without computing any antiderivatives, determine the exact value of

\displaystyle\int_{1}^{5}\bigl(p(x)-q(x)\bigr)\,dx

. Show all reasoning steps. (c) Evaluate

\displaystyle\int_{5}^{1}\bigl(p(x)-q(x)\bigr)\,dx

and indicate whether the result represents a positive, negative, or zero quantity. Justify your answer.