Write the definite integral that represents the area of the region

Question

Use a graphing utility to graph the region bounded by the graphs of the functions. Write the definite integral that represents the area of the region. (Hint: More than one integral may be necessary.)

y = 16x, y = x, x = 1, x = 8

Answer 1

y= 16x

y =x

difference = 16x-x = 15x

integral of 15x = 15x^2/2

evaluated 1 to 8

= 15(8^2)/2 - 15(1^2)/2 = 15(64-1)/2 = 15(63)/2 = 945/2 = 472.5

check the answer with geometry.

draw a sketch, 2 upward sloping lines y=16 and y=x. and 2 vertical lines: x=1 and x=8

calculate the area of the quadrilater formed with those 4 lines, with 4 verticies (1,16), (1,1), (8,128) and (8,8)

Answer 2

Area A = ∫₁⁸15xdx = 15x²/2₁⁸ = 15/2(64 - 1) = 472.5