help? area? curves?

Question

Answer 1

1)

Step 1:

The curve is .

Graph the curve  .

Determine the area below the curve and above the axis.

Observe the graph:

The region shaded in blue colour is the required area of the region.

The points of intersection are and .

Step 2:

Definite integral as area of the region:

If and are continuous and non-negative on the closed interval ,then the area of the region bounded by the graphs of and and the vertical lines and is given by

.

Here and .

Integral limits are and .

Apply power rule of integration: .

Area of the required region is .

Solution:

Area of the required region is .

Answer 2

2)

Step 1:

The curve is .

Graph the curve  .

Determine the area right of the curve and left of the axis.

Observe the graph:

The region shaded in pink colour is the required area of the region.

The points of intersection are and .

Step 2:

Definite integral as area of the region:

If and are continuous and non-negative on the closed interval ,then the area of the region bounded by the graphs of and and the lines and is given by

, .

Here and .

Integral limits are and .

Apply power rule of integration: .

.

Area of the required region is .

Solution:

Area of the required region is .