USE GEOMETRIC FORMULA TO EVALUATE THE INTEGRAL

The graph of f consists of line segments and a semicircle, as shown. Evaluate each definite integral by using geometric formulas.

Problem 1 :

Solution :

The shaded region is a quadrant. The geometric formula to find area of quadrant is (1/4) πr²

The required area is below the x-axis, it should be negative.

radius = 2 units

Area of quadrant = (1/4) πr²

=  (1/4) π(2)²

= -π

Problem 2 :

From the given integral, we understand that the required area is triangle. The shaded region is above the x-axis, it should be positive.

Area of triangle = (1/2) x base x height

base = 4 units and height = 2 units

= (1/2) x 4 x 2

= 4

Problem 3 :

Solution :

From the given definite integral, we understand that the required area will be sum of area of triangle and area of semicircle.

The required region is below the x-axis, so it should be negative.

Area of triangle = (1/2) x base x height

base = 2 and height = 1

= (1/2) x 2 x 1

= 1 ------(1)

Area of semicircle = (1/2) x πr²

radius = 2 units

= (1/2) x π(2)²

= 2π ----(2)

(1) + (2)

 = -(1 + 2π)

Problem 4 :

Solution :

Problem 5 :

Solution :

Problem 6 :

Solution :