Answer to Question #120229 in Calculus for Lissa

Question #120229
Find the area bounded by the curve y = |x − 1|, the x-axis and the lines x = −7 and x =11. (5 marks)
1
Expert's answer
2020-06-07T16:21:08-0400

Let's represent the required area on the coordinate plane:


We can see that it consists of two isosceles right triangles with sides 8 and 10, so we can find the required area as a sum of two areas of triangles.

"A=\\frac{1}{2}*8*8+\\frac{1}{2}*10*10=32+50=82."

Another way to find the area is using integrals. As our area consists of two parts, we have to split integration area in two areas, separated by point x=1.

"y=|x-1|=\\begin{cases} 1-x, & x < 1, \\\\ x-1, & x\\ge 1. \\end{cases}"

"A=\\int_{-7}^{11} |x-1|dx=\\int_{-7}^{1} (1-x)dx+\\int_{1}^{11} (x-1)dx="

"=(x-\\frac{x^2}{2}) |_{-7}^{1} +(\\frac{x^2}{2}-x) |_{1}^{11} ="

"=(1-\\frac{1^2}{2})-(-7-\\frac{(-7)^2}{2})+(\\frac{11^2}{2}-11)-(\\frac{1^2}{2}-1)="

"=1-\\frac{1}{2}+7+\\frac{49}{2}+\\frac{121}{2}-11-\\frac{1}{2}+1=(1+7-11+1)+(-\\frac{1}{2}+\\frac{49}{2}+\\frac{121}{2}-\\frac{1}{2})="

"=-2+\\frac{168}{2}=-2+84=82."

Answer: 82.


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