Question

Find the area of the region bounded by the curve x² = 16y and the line y = 4.

Answer 1

Given:
Curve x² = 16y and the line y = 4

Step 1: Find points of intersection

Put y = 4 in x² = 16y

x² = 16(4) = 64

x = ± 8 

So, the curve and line meet at x = −8 and x = 8.

Step 2: Write the equation of the curve in terms of x.

From x² = 16y,

y = `x^2/16`

Step 3: Area formula

Area between line and curve:

`A = ∫_(−8)^8(4 − x^2/16)dx`

Since the figure is symmetric,

`A = 2 ∫_0^8(4 − x^2/16)dx`

Step 4: Integrate

`A = 2 [4x − 1/16 xx x^3/3]_0^8`

`A = 2[32 − 512/48]`

`A = 2[32 − 32/3]`

A = `2 xx 64/3`

A = `128/3`

Area = `128/3` sq. units