Formal Definition of the Definite Integral
The formal definition of the definite integral is a fundamental concept in calculus, providing a precise mathematical framework for calculating the accumulated total of a function over a specified interval. Developed by mathematicians such as Isaac Newton and Gottfried Wilhelm Leibniz, this definition involves partitioning the interval, evaluating the function at specific points within each subinterval, and taking the limit as the partition becomes infinitesimally small. The result is a powerful tool for analyzing and solving problems related to area, displacement, and other applications in mathematics and physics.
Questions
- How do you find the definite integral of #cot x csc^5 x dx# from #[pi/6, pi/2]#?
- How do you evaluate the definite integral #int sin^(5)x * cos^(20)x dx# from [0,pi/2]?
- How do you find the Improper integral #e^[(-x^2)/2]# from #x=-∞# to #x=∞#?
- How do you find the definite integral for: #sqrt(4+3(t^4))dt # for the intervals #[1, 4]#?
- How do you find the definite integral of #(sqrt(2x+1))dx# from #[1, 4]#?
- How do you find the definite integral of #(3x^3 +7) dx # from #[1, 5]#?
- How do you find the definite integral for: #((2x)/(1+x^2)^(1/3))dx# for the intervals #[0, sqrt7]#?
- What is the integral of dy/dx?
- How do you evaluate the definite integral #int (dx/(xsqrt(lnx)))# from #[1, e^6]#?
- How do you find the definite integral of #2/(sqrt(x) *e^-sqrt(x))# from #[1, 4]#?
- What is #int_(0)^(1) e^(5x^2)dx #?
- How do you find the definite integral for: #x dx# for the intervals #[2, 4]#?
- How do you evaluate the definite integral #int 4/sin^2(z) dz# from #[pi/6, pi/2]#?
- How do you find the definite integral of #sqrt(y+1)*dy# from #[-1, 0]#?
- How do you evaluate the definite integral #int -8/3*2^xdx# from -2 to 9?
- How do you find the definite integral for: #(1/x^(2))dx# for the intervals #[5, 6]#?
- How do you find the definite integral of #dx/ (x+2)# from #[0, 7]#?
- What' the integral of ln √x dx ? (With the substitution)
- How do you find f given #f''(x)=-2+36x-12x^2# and f(0)=2, f'(0)=1?
- How do you find the antiderivative of #f(x)=(x^3-2x^2+x)/x#?