Just diving into integral calculus

In summary, the conversation discusses understanding the area of a shape by using inscribed and circumscribed polygons. The method involves finding the limit as n approaches infinity of the sum of the area of n inscribed rectangles. The conversation also mentions using the limit of a Riemann sum to approximate the area, and dividing the interval into more subintervals for a better approximation. The conversation concludes by discussing the validity of using the function F(x) = (x^2)/2 + x and applying the theorem to find the area of the shape.
  • #1
1MileCrash
1,342
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And the first thing I'm tackling is understanding the area of a shape by inscribed/circumscribed polygons.

For example, the shape produced by the line: y = x + 1, the y axis, and the vertical line x = 2.

Now, my understanding is that the area is the limit as n approaches infinity of the sum of the area of n inscribed rectangles.

Can someone explain how this translates to actual math? How would I find the area of the above shape?
 
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  • #2
Is the other boundary of your region the x axis? If that's the region, its shape is a trapezoid whose area is 4.

Using the limit of a Riemann sum to get the value, you would divide the interval [0, 2] into some number of subintervals, and the use rectangles or some other shape to approximate the area above each subinterval.

For example, if the interval is divided into 4 subintervals, and inscribed rectangles are used, the area estimate is found to be .5*1 + .5*1.5 + .5 * 2.0 + .5 * 2.5 = .5*(7) = 3.5.

To get a better approximation, divide the interval into more subintervals.
 
  • #3
Is the other boundary of your region the x axis? If that's the region, its shape is a trapezoid whose area is 4.

Is it a valid method for me to do F(2) - F(0) to arrive at that answer? Would I be applying the theorem correctly?

[itex]F(x) = \frac{x^2}{2} + x[/itex]

F(2) = 4
F(0) = 0

4-0 = 4

Since we are bounded by x = 0 and x = 2?
 
  • #4
Yes.
 
  • #5
That's really powerful!
 

Related to Just diving into integral calculus

1. What is integral calculus?

Integral calculus is a branch of mathematics that deals with calculating the area under a curve or finding the accumulation of a quantity over an interval. It is used to solve problems involving continuous change and is an essential tool in physics, engineering, economics, and other fields.

2. What is the difference between integral calculus and differential calculus?

Differential calculus focuses on the study of rates of change and slopes of curves, while integral calculus deals with the accumulation of quantities and finding the area under curves. They are often referred to as the two branches of calculus and are used together to solve a wide range of problems.

3. What are the main techniques used in integral calculus?

The main techniques used in integral calculus are integration by substitution, integration by parts, and partial fraction decomposition. These techniques allow us to solve a wide variety of integrals and are crucial in solving real-world problems.

4. How is integral calculus used in the real world?

Integral calculus is used in a variety of fields, including physics, engineering, economics, and finance. It is used to solve problems involving continuous change, such as finding the area under a velocity-time graph to determine displacement, calculating the volume of irregularly shaped objects, and determining the total cost of a production process.

5. What are some common applications of integral calculus?

Some common applications of integral calculus include finding the area under a curve, calculating the volume of a solid, solving optimization problems, determining the center of mass of an object, and analyzing growth and decay processes. It is also used in statistics to calculate probabilities and in signal processing to analyze and filter signals.

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