Have to show area is 24, I get 12?? [Integration Issue]

The question: https://gyazo.com/63d9629626e647a26e5bc5c9ce77bc67
https://gyazo.com/88d2200d199bce319314ddc050d4b80b

What I did:

Found the derivative to in order to find the gradient which was 6
y-15 = 6x-24
y = 6x-9

After rearranging when each point(intercepts) is 0, x = 3/2 and y = -9

Integrated the area of the curve to get 36
Then from that I tried to find the area of the triangle underneath the graph by doing bxh/2:
base is 4
height is 24 (inbetween -9 and 15)
(4x24)/2 is 48 and did 36-48 to get -12 which was just blatantly wrong

A bit hard to read the question, but remember that area below y=0 is negative. This would affect both the curve and the choice of triangle.

Is that 5x^3/2?

Hey I’ve found out the correct way of doing this

Everything you did first was right you have the line of equation y=6x-9

When x= 0 y=-9
When y= 0 x = 1.5

Integrating finds the area between the curve at the x axis therefore the little triangle under the curve has to be calculated separately and added on to the final answer. You do this by doing 9*1.5 / 2, the intersection of x and y axis I stated before. This is 6.75

Then you need to integrate between the limits of 4 and 0 you should get an answer of 36.

Then you take away the triangle under the straight line calculated by 4-1.5 = 2.5
2.5 * 15/ 2 which is 18.75

6.75 + (36-18.75) = 24

Hope this helps let me know if you don’t understand anything

Yikes so the way I did it was purely coincidental that I got 24

But thank you, I did something similar to this but got some crazy answer