Solving 1/2b^2=sin(b): Find b?

  • Thread starter Coriolis314
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In summary, the conversation is discussing the solution for finding the value of b that makes two integrals, R and S, equal to each other. The equation 1/2b^2=sin(b) is mentioned, but it is noted that there may not be an elementary solution and other methods, such as numerical techniques or the Lambert W function, may need to be used. The conversation ends with the acknowledgement that the problem may be meant to be solved using a calculator or an approximation algorithm.
  • #1
Coriolis314
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This is not homework. I ran across this question in a book store a few minutes ago and can't seem to finish it...

R= int(x,0,b
S=int(cos(x),0,b
for what value of b (a positive constant) does R=S?

1/2x^2,0,b=1/2b^2
sin(x),0,b=sin(b)

1/2b^2=sin(b)

Its been a while.. but I should be able to do this... thanks
 
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  • #2
The integrals are easy enough, but I don't think the equation you ended with has an elementary solution. One approach would be to use numeric techniquest to get an approximate solution. Another approach might be the Lambert W function.
 
  • #3
Thank you! I assumed it had a smaller function because it was in a very low level book. I think it is mean to be entered into a calculator at the last point or be cycled through an approximation algorithm.
 

1. What is the purpose of solving the equation 1/2b^2 = sin(b)?

The purpose of solving this equation is to find the value(s) of b that satisfy the equation. This can help in understanding the relationship between 1/2b^2 and sin(b) and can be used in various mathematical and scientific applications.

2. What is the process for solving 1/2b^2 = sin(b)?

The process for solving this equation involves using algebraic principles to manipulate the equation into a form where b can be isolated on one side. This can include multiplying or dividing both sides by a constant, taking the square root, and using trigonometric identities.

3. Can this equation be solved analytically or does it require numerical methods?

This equation can be solved analytically using algebraic manipulation and trigonometric identities. However, for certain values of b, it may be easier or more accurate to use numerical methods such as iteration or approximation techniques.

4. Are there any restrictions on the possible values of b in this equation?

Yes, there are restrictions on the possible values of b in this equation. Since b appears in the denominator of 1/2b^2, the value of b cannot be 0. Additionally, the value of sin(b) can only range from -1 to 1, so the value of b must be within the domain of arcsin function, which is -π/2 to π/2.

5. How many solutions are there for this equation?

The number of solutions for this equation depends on the range of values for b and the intersection of the curves 1/2b^2 and sin(b) on a graph. There could be no solutions, one solution, or multiple solutions. It is also possible for the solutions to be complex numbers rather than real numbers.

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