Find the area of the shaded region in the picture using integration with respect to x.

[giggity]

Find the area of the shaded region in the picture using integration with respect to x.
I can get the answer through integration with respect to y (which is 9 units I believe), just cannot with respect to x. If anyone can post on how to get the answer but with integrating respect to x, that'd be great.
Thanks.

 

[Deleted member 4993]

Find the area of the shaded region in the picture using integration with respect to x.
I can get the answer through integration with respect to y (which is 9 units I believe), just cannot with respect to x. If anyone can post on how to get the answer but with integrating respect to x, that'd be great.
Thanks.
View attachment 21114

While doing integration with 'x' - the lower limit of integration will change for x =0 to x=1 and then x=1 to x= 4.

Do you see why? What will be those lower limits in those domains?

 

[joshuaa]

this is the easy way

[MATH]\int_{-2}^{4} \int_{\frac{y^2}{4}}^{\frac{y + 4}{2}} \ dx \ dy[/MATH]
this is the hard way

[MATH]\int_{0}^{1} \int_{-2\sqrt{x}}^{0} \ dy \ dx + \int_{1}^{2} \int_{2x - 4}^{0} \ dy \ dx + \int_{0}^{2} \int_{0}^{2\sqrt{x}} \ dy \ dx + \int_{2}^{4} \int_{2x - 4}^{2\sqrt{x}} \ dy \ dx[/MATH]

 

[joshuaa]

you can also reduce them to

[MATH]\int_{0}^{1} \int_{-2\sqrt{x}}^{2\sqrt{x}} \ dy \ dx \ + \int_{1}^{4}\int_{2x - 4}^{2\sqrt{x}} \ dy \ dx[/math]