help with integration question?

Feb 2009
101
2
can anyone help me with the process to the question. points i need to research rather than the answers. many thanks

The function \(\displaystyle y=2(x-1)(x-4)^2\)

(a) find the values of A and B
(b) what is the size of the shaded part


 

skeeter

MHF Helper
Jun 2008
16,216
6,764
North Texas
can anyone help me with the process to the question. points i need to research rather than the answers. many thanks

The function \(\displaystyle y=2(x-1)(x-4)^2\)

(a) find the values of A and B
(b) what is the size of the shaded part



should be clear that A = 1 and B = 4 (roots of the cubic, one w/ multiplicity two)

area = \(\displaystyle \int_1^4 2(x-1)(x-4)^2 \, dx\)

expand the cubic, then integrate and evaluate the definite integral using the fundamental theorem of calculus.
 
May 2010
20
8
\(\displaystyle

\)

A and B occur when y=0

\(\displaystyle y=2(x-1)(x-4)^2\)
\(\displaystyle 0=2(x-1)(x-4)^2\)


\(\displaystyle
0=(x-1) \) and \(\displaystyle 0=(x-4)\)

Therefore...
\(\displaystyle A=1\) and \(\displaystyle B=4\)


The Area is..
\(\displaystyle \int2(x-1)(x-4)^2dx\)
\(\displaystyle 2\int(x-1)(x^2-8x+16)dx\)
\(\displaystyle 2\int(x^3-7x^2+24x-16)dx\)

Limits of integration are 1 to 4