Unit #10 De+inite Integration & The Fundamental Theorem Of Calculus


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1 Unit # De+inite Integrtion & The Fundmentl Theorem Of Clculus. Find the re of the shded region ove nd explin the mening of your nswer. (squres re y units) ) The grph to the right is f(x) = x + 8x )Use right hnded Riemnn sums to estimte the re elow the f(x) from [, 8] ) Use sigm nottion nd your clcultor pproximte the re using right hnded rectngles. Pge of 7
2 . Suppose jet plne is trveling with velocity of v(t) = t + 8t (miles/min). At t = the plne is t zero position. ) In reltion to the plne s strting position, wht is the eqution for the plne s position for ny time etween t = nd t = 8. ) Wht is the plne s displcement etween t = to t = 8? Show ll your work. c) Wht is the plne s displcement etween t= to t=8? Show ll your work Is there reltionship etween the re found in question nd the displcement found in question. Riemnn sums cn e royl pin if using lot of rectngles to pproximte the re. But to get n ccurte nswer, numerous rectngles must e used. How else do you think we will determine the re of nonregulr shpes?. Look up either in textook or on the Internet the fundmentl theorem of clculus. Explin how it is similr to wht you did in question nd c. 5. Explin the purpose nd the components of the fundmentl theorem of clculus Pge of 7
3 . Use integrtion to \ind the exct re of the shded region. f(x) = x + x  7. Sketch the region tht represents ech integrl. Use one color for positive re nd nother color for negtive re. 8. If f(x)= x  x + is the function for question 7, determine the resulting nswer fter ech integrl is performed ) ) c) 9. To \ind totl re, you would need to integrte the positive nd negtive regions seprtely nd dd the solute vlues. Find the totl re of the shded region for question 7. Function is still f(x) = x  x + ) ) c) Pge of 7
4 Properties of Integrls I. = II. = III. ( f (x) ± g(x) ) dx = ± g(x)dx IV. k = k if k is constnt numer V. = + if c c < c <. The grph to the right represents the function f(x). Using geometry re formuls determine the following integrls ) ) c) d) e) 8 5 f) Suppose the grph ws velocity of person wlking in stright line. Descrie the person s motion(direction nd mgnitude) for the time t. Mke sketch of the following curve nd the indicted region when the integrl is pplied. Use one color for positive re nd different for negtive re. Then perform the fundmentl theorem, which would let positive nd negtive re s cncel. ) x dx ) x dx c) ( x + )dx d) (x )dx Pge of 7
5 e) (x x)dx f) (x x + 8)dx. This time, \ind the totl re of the region enclosed y the xxis nd the curve y= x x from [,]. (There should not e negtive re). Find the re from the xxis to f (x) = x + from [, ]. perform the integrtion ) x dx ) e x dx c) dx d) x x dx e) (x )dx f ) cos(x) dx π g) dx h) π π sin(x) dx Pge 5 of 7
6 Conceptul mening Explin the mening of the nswer fter the fundmentl theorem is performed. 5. An oject trvels with velocity of v(t) m/sec ; mening of v(t)dt. An oject trvels with velocity of v(t) m/sec ; mening of v(t) dt 7. A tree grows y rte g(t) cm/yer ; mening of 8. An IV drips y the rte s(t) ml/min.; mening of g(t)dt s(t)dt 9. Wter in pool is drining out y the rte of n(t) m /min. A hose llows wter to \low into the pool t the rte h(t) m /min. At t= the pool hs m of wter. ) Wht is the mening of ; ) Wht is the mening of; c) wht is the mening of; d) Wht does it men if h(t)dt n(t)dt is negtive? e) wht does + h(t)dt n(t)dt men?. A ug is crwling ck nd forth in stright line y the rte v(t). ) Wht is the mening of ) Wht is the mening of c) Wht is the mening of d) wht is the mening of n(t)dt h(t)dt h(t)dt v(t)dt v(t)dt n(t)dt v(t)dt + v(t)dt v(t) dt Pge of 7
7 MULTIPLE CHOICE. The grph of piecewise liner function for t is shown to the right. Wht is the vlue of (A) (B).5 (C) (D) 5.5 (E) 8 No clcultor. dx is equl to x (A) / (B) 7/ (C / (D) (E) ln x. sint dt is equl to (A) sin x (B) cos x (C) cos x (D) cos x (E) cos x. The \low of oil in rrels per hour through pipeline on July 9 is given y the grph shown. From the following, which est pproximtes the totl numer of rrels of oil tht pssed through the pipeline tht dy? (A) 5 (B) (C), (D) (E),8 5. Wht re ll vlues of k for which k x dx = (A)  (B) (C) (D)  nd (E) ,, No clcultor. The velocity eqution is v(t)=t + t Wht is the verge velocity from [, ]? (A) (B) 8 (C) (D) (E) Pge 7 of 7
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