# Difference between double and repeated integrals

Hi, I'm just having some trouble with definitions. I've googled repeated integrals but I haven't yet come across something which has answered my question. Anyway, I'd like to know what the difference is, between double and repeated integrals. For example if I had:

$$\int\limits_0^2 {\int\limits_0^x {x^2 y} dydx}$$

I would evaluate it as follows.

$$\int\limits_0^2 {\int\limits_0^x {x^2 y} dydx}$$

$$= \int\limits_0^2 {\left[ {\frac{{x^2 y^2 }}{2}} \right]} _{y = 0}^{y = x} dx$$

$$= \int\limits_0^2 {\left( {\frac{{x^4 }}{2}} \right)} dx$$

$$= \left[ {\frac{{x^5 }}{5}} \right]_0^2$$

= 32/5.

Now I assume that I've just evaluated the integral as a "double integral." The question booklet I have lists the question under "repeated integrals." I'm wondering what the difference is and how I can evaluate this integral as a repeated integral. The only thing I can gather about repeated integrals(from the little bits of info I've found on google) is that the integration is done repeatedly wrt one variable but I'm not sure how that works. Can someone please explain to me how to evaluate this integral as a repeated integral? The answer is 16/5.

Benny said:
Hi, I'm just having some trouble with definitions. I've googled repeated integrals but I haven't yet come across something which has answered my question. Anyway, I'd like to know what the difference is, between double and repeated integrals. For example if I had:

$$\int\limits_0^2 {\int\limits_0^x {x^2 y} dydx}$$

I would evaluate it as follows.

$$\int\limits_0^2 {\int\limits_0^x {x^2 y} dydx}$$

$$= \int\limits_0^2 {\left[ {\frac{{x^2 y^2 }}{2}} \right]} _{y = 0}^{y = x} dx$$

$$= \int\limits_0^2 {\left( {\frac{{x^4 }}{2}} \right)} dx$$

$$= \color{red} \left ( \frac{1}{2} \right ) \cdot \color{black} \left[ {\frac{{x^5 }}{5}} \right]_0^2 \color{red} \ = \ \frac{16}{5}$$

= 32/5.

Now I assume that I've just evaluated the integral as a "double integral." The question booklet I have lists the question under "repeated integrals." I'm wondering what the difference is and how I can evaluate this integral as a repeated integral. The only thing I can gather about repeated integrals(from the little bits of info I've found on google) is that the integration is done repeatedly wrt one variable but I'm not sure how that works. Can someone please explain to me how to evaluate this integral as a repeated integral? The answer is 16/5.
the answer to above integral is 16/5.
you forgot the factor of (1/2) shown in RED above.

Benny said:
Hi, I'm just having some trouble with definitions. I've googled repeated integrals but I haven't yet come across something which has answered my question. Anyway, I'd like to know what the difference is, between double and repeated integrals. For example if I had:

$$\int\limits_0^2 {\int\limits_0^x {x^2 y} dydx}$$

I would evaluate it as follows.

$$\int\limits_0^2 {\int\limits_0^x {x^2 y} dydx}$$

$$= \int\limits_0^2 {\left[ {\frac{{x^2 y^2 }}{2}} \right]} _{y = 0}^{y = x} dx$$

$$= \int\limits_0^2 {\left( {\frac{{x^4 }}{2}} \right)} dx$$

$$= \color{red} \left ( \frac{1}{2} \right ) \cdot \color{black} \left[ {\frac{{x^5 }}{5}} \right]_0^2 \color{red} \ = \ \frac{16}{5}$$

= 32/5.

Now I assume that I've just evaluated the integral as a "double integral." The question booklet I have lists the question under "repeated integrals." I'm wondering what the difference is and how I can evaluate this integral as a repeated integral. The only thing I can gather about repeated integrals(from the little bits of info I've found on google) is that the integration is done repeatedly wrt one variable but I'm not sure how that works. Can someone please explain to me how to evaluate this integral as a repeated integral? The answer is 16/5.
there's no practical difference between a "double integral" and a "repeated integral". the term "repeated integral" generally refers to the method used to evalute a "double integral":

$$\mbox{Double Integral = } \int \, \int_{(2D) Region} f(x,y) \, dA \ = \ \int \, \int_{(x,y) Region} f(x,y) \, dx \, dy \ \mbox{ = Repeated Integral}$$

in your solution above (except for the error shown in RED), you evaluated the "double integral" over a (2D) Region with the method of "repeated integrals" over an (x,y) Region, by first integrating wrt "y" and then integrating wrt "x".

Last edited:
HallsofIvy