Dec 23, 2010 · One is the question of why the definite Riemann integral gives the correct notion of "area under a curve" for a (nonnegative, Riemann integrable) function. The other, which …

Jun 7, 2016 · The definite integral is by definition a limit of a certain class of sums (the class of the Riemann sums, by the way): no relation a priori with the derivatives. The indefinite integral is …

Feb 4, 2018 · The integral of 0 is C, because the derivative of C is zero. Also, it makes sense logically if you recall the fact that the derivative of the function is the function's slope, because …

Dec 4, 2020 · 2 I was finding $$\int_0^\infty e^ {-x} dx$$ So the integral becomes $$\left. (-e)^ {-x}\right\rvert_0^\infty$$ Now here's the confusing thing for me. What exactly does infinity mean …

Jul 2, 2023 · The Riemann integral follows various "intuitive" notions of area, for example if you split an interval up then the integral over the whole interval is equal to the sum of its parts, if …

Apr 2, 2016 · The definite integral is different in the sense that it evaluates either to a number (in the single variable case) or a function independent of the variable of integration (in the …

5 days ago · definite-integrals taylor-expansion estimation numerical-calculus indefinite-integrals calculus integration definite-integrals improper-integrals closed-form definite-integrals closed …

Jun 8, 2015 · The integral we generally teach in a first calculus course actually depends on a parameterization of the interval we are integrating over, and perhaps most naturally …

For an integral of the form $$\tag {1}\int_a^ {g (x)} f (t)\,dt,$$ you would find the derivative using the chain rule. As stated above, the basic differentiation rule for integrals is:

Jun 10, 2012 · A definite integral is nothing different from an indefinite integral but the constant, that was eliminated during the differentiation, has some definite value. For instance in …