How to find the integration bounds when calculating area

To calculate an area between curves, I need to integrate with respect to x between the curve $y=\sqrt{2x}$, the x-axis and the line $y=\frac{4x-12}{5}$

My understanding, using google to display plot is that I need to integrate from 0 to 8 for $y=\sqrt{2x}$ minus the integrate from 3 to 8 for $y=\frac{4x-12}{5}$.

However, I'd like to know if there is any easy and fast method to find integration bounds doins some calculus as I doubt I will have access to google during my math exam ?

You'll need to be able to sketch or analyze functions without Google. :)

You have hints. The curve $y=\sqrt{2x}$ suggests that it isn't straight. At this point, you'd either picture what a square root plot looks like, or graph out a few points on paper to see it.

Where it crosses the $x$ axis is important because that's also one of your bounds. This happens at $y=0$, so it crosses the $x$ axis at the origin.

The line $y=(4x-12)/5$ is just that. Look for the place it crosses the $x$ axis -- again, because that's one of your bounds. This happens at $y=0$, so $x=3$ is the crossing point.

Then, finally see where the line intersects the other curve. You'll have

$$\sqrt{2x} = \frac{4x-12}{5}.$$

Here you can either plot a few points to find that they intersect at $x=8$, or solve the quadratic equation that you get by squaring both sides of the above equation, and picking the root that's in the domain of your problem. (The other root, $9/8$, makes no sense in this context because the line is below the $x$ axis for values of $x$ less than $3$.)

• So there isn't any formula, I'll need to draw each time I want to find my bounds ? – student Dec 17 '14 at 23:48