Area between Curves: Find 1st Quadrant

In summary, the "area between curves" in the first quadrant is the region bounded by two curves in the first quadrant of a coordinate plane. This can be found by calculating the definite integral between the points of intersection of the two curves. The formula for finding this area is A = ∫<sub>a</sub><sup>b</sup> (f(x) - g(x)) dx, and it cannot be negative as it represents net area above the x-axis. Real-life applications include business, gardening, and construction.
  • #1
karush
Gold Member
MHB
3,269
5
Find the area of the region in the first quadrant
bounded by the line $y=x$, the line $x=2$, the curve $y=\frac{1}{x^2}$

$$\int_{1}^{2}\left(x-\frac{1}{{x}^{2}}\right) \,dx$$
just seeing if this is the way to go.

View attachment 4495
 
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  • #2
Looks good to me!
 
  • #3
The rest was easy the Area $ = 1$
 
  • #4
Let's just see about that...

\(\displaystyle \int_1^2\left(x-\frac{1}{{x}^{2}}\right)\,dx=\left[\frac{x^2}{2}+\frac{1}{x}\right]_1^2=\left(2+\frac{1}{2}\right)-\left(\frac{1}{2}+1\right)=1\)

Yep...you are correct. (Mmm)
 

FAQ: Area between Curves: Find 1st Quadrant

What is the "area between curves" in the first quadrant?

The "area between curves" in the first quadrant refers to the region bounded by two curves on a coordinate plane, specifically in the first quadrant. This region can be found by calculating the integral of the two curves between the points of intersection.

How do you find the area between curves in the first quadrant?

To find the area between curves in the first quadrant, you can use the definite integral with the limits of integration being the points of intersection between the two curves. This will give you the area of the region bounded by the two curves.

What is the formula for finding the area between curves in the first quadrant?

The formula for finding the area between curves in the first quadrant is: A = ∫ab (f(x) - g(x)) dx, where f(x) and g(x) are the two curves and a and b are the points of intersection.

Can the area between curves in the first quadrant be negative?

No, the area between curves in the first quadrant cannot be negative. This is because the definite integral calculates the net area, meaning that any areas above the x-axis are considered positive and any areas below the x-axis are considered negative. Since the first quadrant is above the x-axis, the area between curves cannot be negative.

What are some real-life applications of finding the area between curves in the first quadrant?

Finding the area between curves in the first quadrant has many real-life applications, such as calculating the total profit from a business's revenue and cost functions, finding the total area of a garden or field with curved borders, or determining the amount of material needed for a curved surface in construction or manufacturing.

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