# If $\int^2_0 f(x)dx=7$ and $\int^6_2 f(x)dx=15$, what is $\int^6_0 f(x)dx$?

my work is as follows:

$$\int^6_0 f(x)dx = \int^2_0 f(x)dx - \int^6_0 f(x)dx$$

I then got, $$7 - 15 = -8$$

Therefore, $$\int^6_0 f(x)dx = -8$$

I think this is right but I'm not sure, so if someone can confirm or deny, that would be great, thanks!

• Please use MathJax to format. – Saad May 8 '20 at 14:28
• The equality you start with is not true. What did you mean to write? – lulu May 8 '20 at 14:31
• given a<b<c then we have the intuitive formula $\int_a^c f(x) = \int_a^b f(x) + \int_b^c f(x)$ – Aladin May 8 '20 at 14:35
• As a suggestion, it is easy to come up with step functions that meet the assumptions. Say $f(x)=\frac 72$ for $0≤x≤2$ and $f(x)=\frac {15}4$ for $2<x≤6$. You can then work out the answer for this example. That won't prove that the answer is always correct, but you could at least see that $-8$ is not correct. And knowing the correct answer is a great start. – lulu May 8 '20 at 14:36
• Sometimes it can help to think about this in words: "The area under $f$ from $0$ to $2$ is 7, and the area under $f$ from $2$ to $6$ is 15. What's the area under $f$ from $0$ to $6$?" – user113102 May 8 '20 at 14:42

$$\int_0^6f(x)dx=\int_0^2f(x)dx+\int_2^6f(x)dx$$

$$\int_0^6f(x)dx=7+15=22$$

You can also look at what a definite integral means geometrically.

$$\int_a^b f(x)dx$$ is the area under the curve from $$a$$ to $$b$$. For example, for this curve, it would be the blue area.

Now, imagine there is a point $$c$$ between $$a$$ and $$b$$ and you know the area from $$a$$ to $$c$$ and the area from $$c$$ to \$b. For example:

With these two areas, you know that the area from $$a$$ to $$b$$ must be $$22 = 7 + 15$$. Or, to put it in integral terms:

$$\int_a^b f(x) = \int_a^c f(x) + \int_c^b f(x)$$

$$\int_a^c f(x) = 7$$

$$\int_c^b f(x) = 15$$

Therefore

$$\int_a^b f(x) = 7 + 15 = 22$$

In the context of your original question: $$a = 0, c = 2, b = 6$$

Well, it is not hard to notice that when $$0<\text{a}<\text{b}$$:

$$\int_0^\text{b}\text{y}\left(x\right)\space\text{d}x=\int_0^\text{a}\text{y}\left(x\right)\space\text{d}x+\int_\text{a}^\text{b}\text{y}\left(x\right)\space\text{d}x\tag1$$