If the value of the integral (I) is 4, the value of the constant b is ___________ (give answer up to 2 decimal places). $I = \int^\infin_{-\infin} e^{\frac{-x^2}{b}}dx$
Gaussian Integral Formula Recall
The standard form of the Gaussian integral is:
$ \int_{-\infin}^{\infin} e^{-ax^2} dx = \sqrt{\frac{\pi}{a}} $
This formula requires that $a > 0$. The integral $I$ given is:
$ I = \int_{-\infin}^{\infin} e^{\frac{-x^2}{b}} dx $
Relating Given Integral to Standard Form
Comparing the given integral $I$ with the standard form, we can see that:
$ a = \frac{1}{b} $
For the integral to converge, we must have $a > 0$, which implies $\frac{1}{b} > 0$, meaning $b$ must be positive ($b > 0$).
Calculating the Integral Value
Substituting $a = \frac{1}{b}$ into the standard formula, the value of the given integral $I$ is:
$ I = \sqrt{\frac{\pi}{(1/b)}} = \sqrt{\pi b} $
Solving for the Constant b
We are given that the value of the integral $I$ is 4. So, we set the expression for $I$ equal to 4:
$ \sqrt{\pi b} = 4 $
To find $b$, we first square both sides of the equation:
$ (\sqrt{\pi b})^2 = 4^2 $
$ \pi b = 16 $
Now, isolate $b$ by dividing both sides by $\pi$:
$ b = \frac{16}{\pi} $
Numerical Value and Rounding
Using the approximate value of $\pi \approx 3.14159265...$:
$ b \approx \frac{16}{3.14159265} $
$ b \approx 5.092958... $
Rounding the value of $b$ to two decimal places gives:
$ b \approx 5.09 $
Important Questions from Definite Integrals
The Legendre polynomials P n(x), n = 0, 1, 2, ..., satisfying the orthogonailty condition \(\int_{{\rm{ - 1}}}^{\rm{1}} {{{\rm{P}}_{\rm{n}}}\left( {\rm{x}} \right){{\rm{P}}_{\rm{m}}}} \left( {\rm{x}} \right){\rm{dx}}\,{\rm{ = }}\,\frac{{\rm{2}}}{{{\rm{2n + 1}}}}{{\rm{\delta }}_{{\rm{nm}}}}\) on the interval [-1, +1], may be defined by the Rodrigues formula P n(x) = \(\frac{{\rm{1}}}{{{{\rm{2}}^{\rm{n}}}{\rm{n!}}}}\frac{{{{\rm{d}}^{\rm{n}}}}}{{{\rm{d}}{{\rm{x}}^{\rm{n}}}}}{\left( {{{\rm{x}}^{\rm{2}}}{\rm{ - 1}}} \right)^{\rm{n}}}\) . The value of the definite integral \(\int_{{\rm{ - 1}}}^{\rm{1}} {\left( {{\rm{4 + 2x - 3}}{{\rm{x}}^{\rm{2}}}{\rm{ + 4}}{{\rm{x}}^{\rm{3}}}} \right){{\rm{P}}_{\rm{3}}}\left( {\rm{x}} \right){\rm{dx}}} \) is
A parametric curve is defined \(x = cos\left(\frac{\Pi t}{2}\right) , Y= sin\left(\frac{\Pi t}{2}\right)\) in the range of \(0\leq t\leq 1\) . It is rotated about X-axis by 360°.
‘What is the area of the surface generated?
if \(\displaystyle\int\dfrac{\sin x}{\sin (x-a)}dx=Ax+B\log |sin(x-a)|+ C\) where A, B and c are real constants then:
Which of the following is NOT a property of definite integral?
The value of integral \(x\over cos^2 x\) is equal to
