Area between $ 2 y = 4 \sqrt{x}$, $y = 4$, and $2 y + 4 x = 8 $

Question

Sketch the region enclosed by the curves given below. Then find the area of the region.

$ 2 y = 4 \sqrt{x}$, $y = 4$, and $2 y + 4 x = 8 $

Attempt at solution:

I guess I'm supposed to divide the areas into several parts, and then sum up the areas of those parts.

Wolfram alpha shows the sketched area... and I don't think those are the correct sketched areas, because the answer 14 isn't a correct answer.

So can someone tell me which sketched areas am I even looking at?

Answer 1

Simplify your boundary equations:

$$y = 2 \sqrt{x}$$ $$y = 4$$ $$y = -2x +4$$

Sketch the area. You ought to try hand-sketching it to verify.

Split into two double integrals.

$$Area = \int_{x=0}^1\int_{y=lower curve}^{higher curve} dydx+ \int_{x=1}^4\int_{y=lower curve}^{higher curve}dydx$$.

UPDATE/EDIT: You ought to have solved it by now. For reference, here's the full integration and solution.

$$= \int_{x=0}^1\int_{y=-2x+4}^{4} dydx+ \int_{x=1}^4\int_{y=2\sqrt{x}}^{4}dydx$$

$$= \int_{x=0}^1 4 - (-2x+4) dx+ \int_{x=1}^4 4- 2\sqrt{x}\ dx$$

$$= 2\int_{x=0}^1 x\ dx+ \int_{x=1}^4 4- 2\sqrt{x}\ dx$$

$$= x^2 \bigg|_0^1 + 4x \bigg|_1^4 - \frac{4}{3}x^{\frac{3}{2}} \bigg|_1^4 $$

$$= 11/3 $$

Answer 2

For the region where $x$ goes from $x=0$ to $x=1$ $y$ goes from $4-2x$ to $4$.

For the region where $x$ goes from $x=1$ to $x=4$ $y$ goes from $2\sqrt{x}$ to $4$.

Thus, the area is given by

$$\text{Area} =\int_0^1 \int_{4-2x}^4 dydx+\int_1^4 \int_{2\sqrt x}^4 dydx$$

Can you finish?

Answer 3

Here's the region:

You can perform the integration in two parts, where the limits are determined by the values of $x$ where the appropriate two equations have the same value. I'll leave the details of that exercise to you but the result is ${11 \over 3}$.

Since you asked, here's the Mathematica code:

Plot[{4, 2 Sqrt[x], If[0 < x < 1, 4, Null], If[1 < x < 4, 4, Null], 
  4 - 2 x}, {x, -1, 5},
 PlotStyle -> {Red, Green, Red, Red, Orange},
 Filling -> {
   {3 -> {{5}, {None, Green}}}, 
   {4 -> {{2}, {None, Blue}}}
   }]