Integration by finding limits?

In summary, the conversation is about a person struggling with integration and seeking help with finding the area under a curve using certain limits and the Fundamental Theorem of Calculus. They are able to solve the problem using the Fundamental Theorem of Calculus, but are having trouble with finding limits. They are unsure if they should use Riemann sums or another method to find the limit.
  • #1
Ruckstar033
1
0
Integration by finding limits?

Hi Guys, I am having a huge problem with integration at the moment and don't know how to approach it cause i have a lousy teacher who couldn't be bothered in actually doing examples on the board. The equation is given as this:

f(t) = 2^t + t^2

Using the equation find the area under the curve between 0 and 1 using:

1) finding certain limits and

2) Fundamental theorem of calculus.

I know how to do 2 as integrating the equation yields 2^t/ln2 + t^3/3 and then sub in the numbers for 1 and 0 and subtract.

Problem is i can't do part 1! Can someone help me with this I am desperate. My test is tomorrow! Your contribution is much appreciated. Thanks
 
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  • #2


Riemann sum?
 
  • #3


[tex]\int_a^b{f(x)}\, \text{d}x = \lim_{n \to \infty}{\sum_{i=1}^n{f(x_i)} \Delta x[/tex]

Is the definition of an definite integral. If she said to find the limit then it seems like she doesn't want you to estimate with a large n value, but to actually find the limit of the sum.
 

What is integration by finding limits?

Integration by finding limits is a method of calculating the area under a curve by using the concept of limits. It involves breaking up the area into smaller, simpler shapes and finding the limit of the sum of these shapes as they approach infinity.

Why is integration by finding limits useful?

Integration by finding limits is useful because it allows us to calculate the area under curves that cannot be easily integrated using traditional methods. It is also used in various fields of science, such as physics and engineering, to solve complex problems.

What are the steps involved in integration by finding limits?

The first step is to identify the function and the limits of integration. Then, the function is divided into smaller, simpler shapes such as rectangles or trapezoids. The limit of the sum of these shapes is then calculated as they approach infinity. Finally, the limit is evaluated to find the area under the curve.

What are some common applications of integration by finding limits?

Integration by finding limits is commonly used in physics to calculate the work done by a variable force and in economics to determine the total revenue or cost of a business. It is also used in engineering to find the total displacement of an object under a varying force.

What are some limitations of integration by finding limits?

Integration by finding limits can be a time-consuming process, especially if the function is complex. It also requires a good understanding of limits and may not always provide an exact solution. Additionally, it may not be applicable to all types of functions.

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