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Definite Integral

Problems

Properties of the Definite Integral

Antiderivatives and the Fundamental Theorem of Calculus

Problem : Given that e x dx = e 2 - and e x dx = 1 - , find e x dx .

By telescoping limits, we have

e x dx + e x dx = e x dx    

so

e x dx = e 2 - - 1 - = e 2 - 1.    

Problem : If xdx = 3/2 , x 2 dx = 7/3 , and (x 2 + 2x)dx = 180/3 , find (x 2 + 2x)dx .

Here we use all three properties introduced in this section:


x 2 + 2xdx = x 2 dx + 2 xdx + x 2 + 2xdx  
  = +2 +  
  =  

Problem : Compute + 3 dx using elementary geometry.

We have


+ 3dx = dx + 3dx  
  = + 6  

since the two integrals correspond respectively to a semicircle of radius 1 and a rectangle with height 3 and width 2 .

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