Area Between Curves

Question

Find the area of the region in the first quadrant bounded by the line y=x, the line x=2, the curve y=1/x^2, and the x-axis. Draw the graph with the area of the region aswell.

Answer 1

The area is the integral from 0 to 1 of y = x plus the integral from 1 to 2 of 1/x^2

A = ₀∫¹x dx + ₁∫² x^-2 dx

A = (1/2)x²]evaluated between 0 and 1 + -x^-1]evaluated between 1 and 2

A = (1/2)1² - (1/2)0² + -(2^-1) - -(1^-1)

A = 1/2 - 0 - 1/2 + 1

A = 1 square unit

Answer 2

Use Desmos to see the graphs: y = x, x = 2, y = 1/x²

Then, you will see the area between all the functions: looks like kinda sector little bit....

y = x is above the y = 1/x².

and they intersect at 1 => x = 1/x² => x^3 = 1 => x = 1

Hence,

S(1,2) (x - 1/x²) dx = 1/2 * x² - (-1/x) ] (1,2)

= 1/2 * (4) + 1/2 - (1/2 + 1) = 1