I'm in a bit of a dilemma here. I'm writing my IB maths IA and I've come across the following definite integral which I just can't figure out
$$ \int_{-0.4}^{0.6}(8x^3+7.5x^2-0.5x+2)^{1/2}dx $$
This comes from the arc-length formula of a quartic polynomial curve. Is there any efficient way of solving this by hand? IB does not like the use of technology to solve maths...
I'm not that skilled in Calculus (yet), and I'm wondering if it is possible to solve, or even approximate, integrals of the square root of high degree polynomials. I'd be content with a semi-accurate approximation that can be done reasonably, say in 1/2 page.
What's the best way to solve these integrals without relying on technology? I've done some reading, but it seems that all the example problems use simpler quadratic polynomials...
Any help or insight appreciated! Thanks
Your integrand does not change much over the region of integration, so just doing a five point trapezoidal integration should be quite accurate. Are you at least allowed a calculator to compute the values? Your integral is approximately $(f(-0.4)+2f(-0.2)+2f(0)+2f(0.2)+2f(0.4)+f(0.6))/10$. Alpha reports the correct value is about $1.66756$ and does not give an exact form, so I doubt one can integrate it analytically.