Area Under Graph: Calculate (1+2^x) from 0 to 30

[whatdofisheat]

to take the area under a graph of
(1+2^x) from 0 to 30 how can you do this and what is the answer

 

[bomba923]

$$\int\limits_0^{30} {\left( {1 + 2^x } \right)dx} = \left( {x + \frac{{2^x }}{{\log 2}}} \right)_0^{30} = 30 + \frac{{2^{30} - 1}}{{\log 2}}$$

Questions like these belong here :)

 

[quasar987]

If you have a function f(x), it's integral from a to b is the area bounded by the graph of the function, the x-axis and the lines y=a, y=b.

The term "area under the curve" can lead to confusion, because in the intervals where the function is negative, the integral actually gives the negative of the area not under but above the curve.

exemple: the integral of sin(x) between 0 and 2pi is 0 because there is a "mountain" over the x-axis, followed by a "valley" (under the x-axis). The total area under the mountain is A, and the area over the valley is A also, but since f(x) is negative there, this area is substracted while calculating the integral...

$$\int_0^{2\pi}\sin(x)dx = \int_0^{\pi}\sin(x)dx+\int_{\pi}^{2\pi}\sin(x)dx = A+(-A)=0$$

 

[bomba923]

The term "area under the curve" can lead to confusion, because in the intervals where the function is negative, the integral actually gives the negative of the area not under but above the curve.

True indeed, but how does that apply here? 1+2^x > 0 for all real x, so $\int\limits_0^{30} {\left( {1 + 2x} \right)dx}$ is (represents) indeed, as asked by the OP, the area under the curve y=1+2^x (additionally bounded by y=0, x=0, and x=30).

to take the area under a graph of
(1+2^x) from 0 to 30 how can you do this and what is the answer

Hint:
$$\forall a > 0,\;\int {a^x dx} = \frac{{a^x }}{{\log a}} + C$$

 

[HallsofIvy]

bomba923, when you delete a message do you have a "Physically remove" button below "Delete" and "Do Not Delete"? If so, clicking on that will not leave the "debris" behind that just using "Delete" does!

(I tried sending this as a private message but your box is full. Please go to your User CP and delete at least some of the private messages you have been getting.)

 

[bomba923]

bomba923, when you delete a message do you have a "Physically remove" button below "Delete" and "Do Not Delete"? If so, clicking on that will not leave the "debris" behind that just using "Delete" does!

I don't have a "physically remove" button anywhere!

*Can you add it to my account?
(*e.g., somewhat like adding avatar privileges to PF contributors' accounts, but in this case, adding a button/option around the "delete post" button...something like that, so I can physically my remove posts )