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Calculating Area with Upper and Lower Limits

Learn how to find the area enclosed by curves using upper and lower limits in calculus. Work through examples step by step to understand the process.

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Calculating Area with Upper and Lower Limits

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  1. Page 253 Example 1 Red line = ”over function” = 4 – 3x. Blue line = ”under function” = 3x2 – 3x + 1. Work out 1 – 2 = 4 – 3x – (3x2 – 3x + 1) = 3 – 3x2 Work out primitive function for 3 – 3x2. This is F(x) = 3x – x3 Work outF(x) for x=1. This is 3.1 – 13 = 2. Call this value1. Work outF(x) for X=-1. This is 3.-1 – (-1)3. = -2 Call this value2. Value1 – value2 = 2 – (-2) = 4. Answer = 4. Area

  2. Area Beräkna arean av det område som begränsas av kurvorna Y=X2-4 och Y=2X-X2. Page 254. ”Over function” is Y=2X – X2. ”Under function” is Y=X2 – 4. Work out Y=2X – X2 – (X2 – 4). This gives Y=2X-2X2+4. Primitive function F(X) = x2 – 2x3/3 + 4X This time wehave to work out the upper and lower limits as we are not given them IN THE EXAMPLE. Thereforesolve X2-4=2X-X2. 2X2-2x-4 = 0  X2-x-2 = 0  (x-2)(x+1)=0 X1 = 2 and x2 = -1

  3. Area Primitive function F(X) = x2 – 2x3/3 + 4X X1 = 2 and x2 = -1 Work out F(X) for X=2. This gives 4 – 2.8/3 + 8 = 20/3. Work out F(X) for X=-1. This gives 1 – (2.-13)/3 + 4.-1 = 1+(2/3)-4. This gives 2/3 – 3 = -7/3. Answer= F(2) – F(-1) = 9 Note: Ifthere is no ”over function” and ”under function” pleasesee page 258

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