Solving an Intriguing Integral: x-1 in the First Quadrant

  • Thread starter Knissp
  • Start date
  • Tags
    Integral
In summary, an integral is a mathematical concept that represents the area under a curve in a graph. Solving an integral means finding the function that represents this area, which involves finding the constant of integration and evaluating the integral at specific limits. The integral of x-1 in the first quadrant is intriguing because it involves finding the area under a curve that is both positive and negative. To solve this integral, you can use symmetry, substitution, or integration by parts. Applications of solving this type of integral include calculating displacement, velocity, and acceleration, as well as profit and loss in business, and other scientific and technological areas.
  • #1
Knissp
75
0

Homework Statement


[tex]\int\int_R x-1 dA[/tex]
R is the region in the first quadrant enclosed between y=x and y=x^3



Homework Equations





The Attempt at a Solution



I set up the bounds as follows: [tex]\int_{x=0}^1\int_{y=x^3}^x x-1 dydx[/tex]

Integrating, I get -7/60, verified with CAS.

I thought this was an easy problem but my answer doesn't match the textbook (-1/2 but could be a misprint, right?) or did I somehow put the bounds of integration wrong?
 
Physics news on Phys.org
  • #2
It looks fine to me.
 
  • #3
I agree with your answer
 

FAQ: Solving an Intriguing Integral: x-1 in the First Quadrant

1. What is an integral?

An integral is a mathematical concept that represents the area under a curve in a graph. It is used to find the total value of a function over a given interval.

2. What does it mean to solve an integral?

To solve an integral means to find the function that represents the area under a curve in a graph. This involves finding the value of the constant of integration and evaluating the integral at specific limits.

3. Why is the integral of x-1 in the first quadrant intriguing?

The integral of x-1 in the first quadrant is intriguing because it involves finding the area under a curve that is both positive and negative. This requires a different approach compared to finding the area under a curve that is entirely positive.

4. How do you solve the integral of x-1 in the first quadrant?

To solve the integral of x-1 in the first quadrant, you can use the concept of symmetry to break the integral into two parts and then evaluate each part separately. You can also use substitution or integration by parts to solve the integral.

5. What are the applications of solving an intriguing integral like x-1 in the first quadrant?

Solving an intriguing integral like x-1 in the first quadrant has various applications in physics, engineering, and economics. It is used to find the total displacement, velocity, and acceleration of an object, as well as the total profit or loss in a business. It is also used in other areas of science and technology that involve finding the total value of a function over an interval.

Similar threads

Probability that 𝑌>3𝑋 where 𝑋,𝑌 are 𝑁(0,1) random variables
Replies
9
Views
1K
Calculate this Integral around the Circular Path using Green's Theorem
Replies
3
Views
1K
Line integral of a scalar function about a quadrant
Replies
4
Views
1K
Volume of Static Solid Using Cross-Sectional Area (Integration)
Replies
2
Views
551
Verify Green's Theorem in the given problem
Replies
10
Views
1K
Stick is broken randomly EDIT TWICE, what is expected length of small side?
Replies
29
Views
2K
Please evaluate this double integral over rectangular bounds
Replies
10
Views
2K
Need help deducing the region for this double integration problem
Replies
9
Views
1K
How do I evaluate this integral?
Replies
6
Views
1K
Double Integral: How to Evaluate a Double Integral over a Pentagonal Region
Replies
2
Views
1K

Hot Threads

Recent Insights

Back
Top