- #1
jaychay
- 58
- 0
Can you please check it for me that I have done it wrong or not ?
Thank you in advance.
Calculating Area Under a Curve: Is My Approach Correct?
In summary, the conversation involves checking for errors in multiple integrals and simplifying them. The first integral can be split into two parts, while the second one involves the secant function. The second problem requires using substitution to simplify the integral.
Can you please check it for me that I have done it wrong or not ?
Thank you in advance.
- #2
skeeter
- 1,103
- 1
w/ respect to x, top function - bottom function
$\displaystyle \int_{-1}^0 9 - 9^{-x} \, dx + \int_0^2 9 - 3^x \, dx$
integral w/respect to y is ok and it can be simplified ...
$\displaystyle \dfrac{3}{2\ln{3}} \int_1^9 \ln{y} \, dy$
secant function integral is set up correctly ... antiderivative is rather easy to see if you recognize the chain rule
$\displaystyle \int_{-1}^0 9 - 9^{-x} \, dx + \int_0^2 9 - 3^x \, dx$
integral w/respect to y is ok and it can be simplified ...
$\displaystyle \dfrac{3}{2\ln{3}} \int_1^9 \ln{y} \, dy$
secant function integral is set up correctly ... antiderivative is rather easy to see if you recognize the chain rule
- #3
HOI
- 921
- 2
The second problem looks pretty standard- take x from 0 to $\sqrt{\frac{\pi}{3}}$ and, for each x, y from 0 to $2xsec^2(x^2)$.
The area is $\int_0^{\sqrt{\frac{\pi}{3}}} 2x sec^2(x^2)dx$.
Let $u= x^2$ so that $du= 2xdx$. When x= 0, u=0 and when x= $\sqrt{\frac{\pi}{3}}$, $u= \frac{\pi}{3}$.
The integral becomes $\int_0^{\frac{\pi}{3}} sec^2(u)du$.
The area is $\int_0^{\sqrt{\frac{\pi}{3}}} 2x sec^2(x^2)dx$.
Let $u= x^2$ so that $du= 2xdx$. When x= 0, u=0 and when x= $\sqrt{\frac{\pi}{3}}$, $u= \frac{\pi}{3}$.
The integral becomes $\int_0^{\frac{\pi}{3}} sec^2(u)du$.
FAQ: Calculating Area Under a Curve: Is My Approach Correct?
What does "finding the area under the curve" mean?
Finding the area under the curve refers to calculating the total area that is enclosed by a curve on a graph and the x-axis. It is often used to determine the total quantity or value represented by the curve.
Why is finding the area under the curve important?
Finding the area under the curve is important in many scientific and mathematical applications. It can provide valuable information about the behavior and characteristics of a system or process, and can also be used to make predictions and solve problems.
How is the area under the curve calculated?
The area under the curve is calculated using integration, which is a mathematical process that involves finding the sum of infinitely small rectangles under the curve. This can be done using various methods, such as the Riemann sum or the trapezoidal rule.
What are some real-world examples of finding the area under the curve?
Finding the area under the curve can be applied in various fields such as physics, economics, and biology. For example, it can be used to calculate the total distance traveled by an object over time, the total revenue generated by a company over a period of time, or the total population of a species over a certain area.
Are there any limitations to finding the area under the curve?
There are certain limitations to finding the area under the curve, such as the accuracy of the data and the assumptions made in the calculation. It is also important to consider the limitations of the chosen integration method and the complexity of the curve being analyzed.