Desperately need help with integration and area

Question

Find the area enclosed between the curve y = 4x^3 - 12x^2 - 40x  and the x-axis

 

What do you do if it just says and the x -axis? i know what to do when normal values are given

 

 

Answer 1

The curve y = 4x³ - 12x²  - 40x

The curve crosses the x axis when y = 0.

 4x(x² - 3x - 10) = 0

 4x(x² - 5x + 2x - 10) = 0

 4x[ x(x - 5) + 2( x - 5)] = 0

 4x(x - 5)(x + 2) = 0

So the limits are x = 0, x = 5 and x = - 2.

We calculate the area A between the curve, x axis and the ordinates first x = - 2 to x = 0.

Next we need to calculate the area between the curve,x axis and the ordinates x = 0 to x = 5.

Total area = A₁  + A₂

Area = 32 - 375 = - 343 square units.