MHB Tom's question at Yahoo Answers regarding solving for a limit of integration

  • Thread starter Thread starter MarkFL
  • Start date Start date
  • Tags Tags
    Integration Limit
AI Thread Summary
To solve the integral from 0 to x of the function 5000(1 - 100/(t + 10)^2) dt, the integral can be expressed as I. By applying the Fundamental Theorem of Calculus, the equation simplifies to a quadratic form, leading to the expression 5000x^2 - Ix - 10I = 0. The quadratic formula can then be used to find x, with the requirement that I equals 25000 for x to equal 10, while also yielding a second solution of x = -5. The discussion encourages further calculus questions to enhance understanding.
MarkFL
Gold Member
MHB
Messages
13,284
Reaction score
12
Mathematics news on Phys.org
Hello Tom,

Since I don't know the value the definite integral is to have, I will use $I$:

$$5000\int_0^x 1-\frac{100}{(t+10)^2}\,dt=I$$

First, let's divide through by 5000:

$$\int_0^x 1-\frac{100}{(t+10)^2}\,dt=\frac{I}{5000}$$

Next, let's use the anti-derivative form of the FTOC on the left side:

$$\left[t+\frac{100}{t+10} \right]_0^x=\frac{I}{5000}$$

$$\left(x+\frac{100}{x+10} \right)-\left(0+\frac{100}{0+10} \right)=\frac{I}{5000}$$

$$x+\frac{100}{x+10}-10-\frac{I}{5000}=0$$

Now, multiply through by $x+10$:

$$x(x+10)+100-\left(10+\frac{I}{5000} \right)(x+10)=0$$

Arrange in standard quadratic form:

$$x^2+10x+100-10x-100-\frac{I}{5000}x-\frac{I}{500}=0$$

$$5000x^2-Ix-10I=0$$

Applying the quadratic formula, we find:

$$x=\frac{I\pm\sqrt{I^2+200000I}}{10000}$$

Now, you just need to substitute the value of $I$ to find the two possible values of $x$, taking care not to cross the singularity in the original integrand.

In order for $x=10$, we find that we require $$I=25000$$, however, this also allows $x=-5$.

To Tom and any other guests viewing this topic, I invite and encourage you to post other calculus questions here in our http://www.mathhelpboards.com/f10/ forum.

Best Regards,

Mark.
 
Suppose ,instead of the usual x,y coordinate system with an I basis vector along the x -axis and a corresponding j basis vector along the y-axis we instead have a different pair of basis vectors ,call them e and f along their respective axes. I have seen that this is an important subject in maths My question is what physical applications does such a model apply to? I am asking here because I have devoted quite a lot of time in the past to understanding convectors and the dual...
Fermat's Last Theorem has long been one of the most famous mathematical problems, and is now one of the most famous theorems. It simply states that the equation $$ a^n+b^n=c^n $$ has no solutions with positive integers if ##n>2.## It was named after Pierre de Fermat (1607-1665). The problem itself stems from the book Arithmetica by Diophantus of Alexandria. It gained popularity because Fermat noted in his copy "Cubum autem in duos cubos, aut quadratoquadratum in duos quadratoquadratos, et...
Insights auto threads is broken atm, so I'm manually creating these for new Insight articles. In Dirac’s Principles of Quantum Mechanics published in 1930 he introduced a “convenient notation” he referred to as a “delta function” which he treated as a continuum analog to the discrete Kronecker delta. The Kronecker delta is simply the indexed components of the identity operator in matrix algebra Source: https://www.physicsforums.com/insights/what-exactly-is-diracs-delta-function/ by...
Back
Top