Question

Find the area of the region bounded by the given curves.y = 6x^2 ln x, y = 24 ln x?

Answer 1

y = 6x²log x, y = 24 log x

24 log x - 6x²log x = 0

Multiply each side by negative one.

6x²log x - 24 log x = 0

Take out common term log x.

log x(6x² - 24) = 0

log x = 0 and (6x² - 24) = 0

log x = 0 then x = e⁰ ⇒ x = 0

6x² - 24 = 0

Add 24 to each side.

6x² = 24

Divide each side by 6.

x² = 24 ⇒ x = ±2.

The two equation interval is (1, 2, -2)

In interval 1<x<2, y₂>y₁ which makes enclosed area A.

Top minus bottom

integrals with logarithms

.

Therefore The area is -9.702.

Comments

= 32 ln 2 - 24 + 14/3                 (where, ln is the natutal logerthem)

= 32(0.69315) - 24 + 4.66

= 22.18 - 19.34

= 2.85.

The area of the region bounded by the curves y = 6x^ ln x and y = 24 ln x is 2.85 square units.