Answer
A) 4
B) $-2\pi$
C) $\frac{9}{2}$$-2\pi$
Work Step by Step
Like we did in question 33, using the definition discussed in Section 4.1, we know that the area under the curve of the function and above the x-axis represents the result of the integral, also if there is area below the x-axis, it is subtracted rather than added from the total sum of areas.
It follows that, to evaluate the integral it suffices to use geometric approach to find the area.
A) The curve under the the function is a right-angle triangle of 2x4 dimensions, so the integral equals the total area which is : $0.5*2*4 = 4$.
B) The evaluation of the integral requires the area of a semi-circle with a radius of 2 units.
Evaluated Integral = $-$ total area = ($-$) 0.5*$2x^{2}$ * $\pi$ = $- 2\pi$.
C) Using part A and part B and the area of the last part which is 0.5*1*1 = 0.5
Total area = 4 $-2\pi$ +.5 >> $\frac{9}{2} -2\pi$.
