This preview shows page 1. Sign up to view the full content.
Unformatted text preview: Areas & Distances
The area problem:
What's the area of the shaded region: We can approximate the area with a set of rectangles. The more rectangles, the better the approximation. These four give a very rough approximation: These eight give a better approximation: These 16 do still better: MTH 173 section 5.1 notes page 1 This many (very thin) rectangles give quite a good approximation: And still better with even more: Then so many rectangles that the sides seem to coincide: MTH 173 section 5.1 notes page 2 2. This problem asks that the area under the curve be approximated with six rectangles in different ways: using left endpoints for height: using right endpoints for height: using midpoints for height: defn: The area of the region that lies under the graph of a continuous function the limit of the sum of the areas of the approximating rectangles lim lim lim is MTH 173 section 5.1 notes page 3 18. Area expression for ln ln lim ln 20 given: lim determine region with this area here so , so ! " to The interval is from The function is Thus the summation given will produce the area under the curve # from to MTH 173 section 5.1 notes page 4 The distance problem Given a changing speed, how far is travelled in time $ ? Here's what Galileo did 400 years ago, before calculus: If we graph constant velocity % against time $, where we all know that & graph: %$. Then in the velocity axis v time axis t1 The area of the rectangle represents distance. Galileo had proposed that for a free falling object, its velocity % after $ seconds was % '$, where ' was a constant Then Galileo thinks of the graph of % mathematics. '$ and area of a geometric shape - the language of Next he progresses to what these days we call a Riemann sum - he imagines a series of short interval constant velocity motions and adds the areas of the resulting rectangles - several rectangles many rectangles
MTH 173 section 5.1 notes page 5 He concluded that if % '$ were true, then distance ( triangle $ '$ '$ We know that '$ ($ '$ the area of the Is there a connection between antiderivatives and areas? Beats me. I sure don't know. 16. Given this velocity graph, estimate the distance travelled in 30 seconds: I used 5 second intervals and the velocities shown in the graph below. The velocities were converted to km/s by dividing by 3600. distance km MTH 173 section 5.1 notes page 6 12. $ % ( ft a. b. ( ft MTH 173 section 5.1 notes page 7 ...
View Full Document