1.6　微积分基本定理

课时过关·能力提升

基础巩固

1下列定积分的值等于1的是(　　) 学

A.∫_0^1▒ xdx B.∫_0^1▒ (x+1)dx

C.∫_0^1▒ 1dx D.∫_0^1▒ 1/2dx

解析∫_0^1▒ 1dx=x"|" _0^1=1-0=1,故选C.

答案C

2∫_2^4▒ 1/xdx的值为(　　)

A.-2ln 2 B.2ln 2

C.-ln 2 D.ln 2

解析∫_2^4▒ 1/xdx=ln x"|" _2^4=ln 2.

答案D

3∫_2^4▒ (x3+x2-30)dx的值为(　　)

A.56 B.28 C.14 D.56/3

解析∫_2^4▒ (x3+x2-30)dx=(1/4 x^4+1/3 x^3 "-" 30x) "|" _2^4=1/4(44-24)+1/3(43-23)-30×(4-2)=56/3.故选D.

答案D

4若∫_0^T▒ x2dx=9,则常数T的值为　　　　　.

解析∵∫_0^T▒ x2dx=x^3/3 "|" _0^T=T^3/3=9,∴T=3.

答案3

5若∫_0^1▒ (2xk+1)dx=2,则k=　　　　　.

答案1

6若f(x)={■(x^2 "," 0≤x≤1"," @2"-" x"," 1

解析∫_0^2▒ f(x)dx=∫_0^1▒ x2dx+∫_1^2▒ (2-x)dx=x^3/3 "|" _0^1+(2x"-" x^2/2) "|" _1^2=1/3+[(2×2"-" 2^2/2)"-" (2"-" 1/2)]=5/6. 学 ]

答案5/6

7如图,曲线y=cos x与直线x=-π,x=3/4π,y=0所围成图形的面积S=　　　　　.