# L33 - L33 – Indeﬁnite Integrals and Total Change...

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Unformatted text preview: L33 – Indeﬁnite Integrals and Total Change Theorem x FTC I – If f is continuous, then a f (t) dt is b FTC II – a f (x) dx = where Indeﬁnite Integrals Deﬁnite Integrals 1 cf (x) dx = k dx = [ f (x) ± g (x) ] dx = 1 dx = x ax dx = xn dx = ex dx = sin(x) dx = sec2 (x) dx = cos(x) dx = csc2 (x) dx = sec(x) tan(x) dx = 1 dx = 1 − x2 csc(x) cot(x) dx = 1 dx = 1 + x2 √ 2 Evaluate: √ 1+ x √ dx x π 4 0 sin(θ) dθ cos2 (θ) tan2 (x) dx 3 (y − 12 ) dy y ( 2x + 4 − ex ) d x 1 + x2 2 1 1 (2 + x)3 dx x 4 Total Change Theorem The integral of a rate of change of a function is the total change of the function on the interval b F (x) dx = a If the volume of water in a lake is increasing at the rate V (t), then t2 V (t) dt = t1 gives 5 If a population is growing at a rate of t2 t1 dn , then dt dn dt = dt gives If the mass of a rod from the left end to point x is given by m(x) the linear density is m (x) = ρ(x), then b ρ(x) dx = a 6 Application – Suppose a particle is moving along a straight line with position function s(t), velocity v (t), and acceleration a(t) . t2 v (t) dt = t1 t2 a(t) dt = t1 total distance travelled= 7 A particle moves along a line so that its velocity at time t is v (t) = t2 − 2t − 8 m/sec a) Find the displacement during the time 2 ≤ t ≤ 5 . b) Find the total distance travelled during the time 2 ≤ t ≤ 5 . 8 Suppose a population of birds is increasing at the rate of 100+20t per year. What is the total increase in population between the sixth and eight years? 9 ...
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L33 - L33 – Indeﬁnite Integrals and Total Change...

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