Question

I have an integral that has to endpoints that are the same. Does that mean that the answer would be zero?

Answer 1

\[ \int_a^a f(x) \ dx = 0 \] for all functions f. It makes sense as the area under the curve is nothing.

Answer 2

\[\int\limits_{3}^{3}(\cos^2(2x)+\sqrt{x})dx\]

Answer 3

(there's one technical exception you learn about in second year university, but ignore that for now. Certainly for continuous functions--such as the one you just typed--this is absolutely correct.)

Answer 4

that is what i thought... thank you...

Answer 5

what is your lower endpoint is a higher value than your upper endpoint

Answer 6

that is supposed to be what if

Answer 7

\[ \int_a^b f = -\int_b^a f \]

So you can 'fix' the order this way.

If a > b and f(x) > 0 for all x in [a,b], then
\[ \int_a^b f \ < 0 \]

Answer 8

For example
\[ \int_{100}^1 x^2 \ dx < 0 \]

Answer 9

so your answer is just nega

Answer 10

negated

Answer 11

\[ \int_{100}^1 x^2 \ dx = -\int_1^{100} x^2 \ dx = .... \]

Answer 12

so for the integral \[\int\limits_{-1}^{3}4x^3dx \] the answer is 80... but you would negate that if your integral began with the lower value on the top?

Answer 13

yes,
\[ \int_{-1}^3 4x^3 \ dx = - \int_3^{-1} 4x^3 \ dx \]

Answer 14

thank you!!