diff --git a/2bc0c8-2024_spring_TA/main.typ b/2bc0c8-2024_spring_TA/main.typ index 26047e5..b6f88c0 100644 --- a/2bc0c8-2024_spring_TA/main.typ +++ b/2bc0c8-2024_spring_TA/main.typ @@ -648,9 +648,9 @@ $ &quad integral.double_D y / (1+x^2+y^2)^(3 / 2) dif x dif y quad D=[0,1]times[0,1]\ &=integral_0^1 dif x integral_0^1 y / (1+x^2+y^2)^(3 / 2) dif y\ &=integral_0^1 dif x integral_0^1 (1 / 2 dif y^2) / (1+x^2+y^2)^(3 / 2)\ -&=integral_0^1 -(1+x^2+y^2)^(-1 / 2)|_0^1 dif x\ +&=integral_0^1 [-(1+x^2+y^2)^(-1 / 2)]_0^1 dif x\ &=integral_0^1 -1 / sqrt(2+x^2) + 1 / sqrt(1+x^2) dif x\ -&=-ln(x+sqrt(2+x^2)) + ln(x+sqrt(1+x^2))|_0^1\ +&=[-ln(x+sqrt(2+x^2)) + ln(x+sqrt(1+x^2))]_0^1\ &=-ln(1+sqrt(3)) + ln(1+sqrt(2)) + 1 / 2ln 2\ &=quad ln(-1+sqrt(3))+ln(1+sqrt(2))-1 / 2ln 2 $ @@ -693,7 +693,7 @@ $ $ &quad integral.double_D x^2 / y^2 dif x dif y\ &= integral_1^2dif x integral_(1 / x)^x x^2 / y^2 dif y\ -&= integral_1^2 (-x^2 / y)|_(y=1 / x)^(y=x) dif x\ +&= integral_1^2 [-x^2 / y]_(y=1 / x)^(y=x) dif x\ &= integral_1^2 (-x+x^3) dif x\ &= -2^2 / 2 + 2^4 / 4 + 1 / 2 - 1 / 4\ &= 9 / 4 @@ -800,7 +800,7 @@ $ &=integral_0^(arctan R) dif theta integral_0^R sec^2 theta dot r dif r\ // &=integral_0^(arctan R) dif theta sec^2 theta r^2/2 |_0^R\ &=R^2/2 integral_0^(arctan R) sec^2 theta dif theta\ -&=R^2/2 tan theta |_0^(arctan R)\ +&=R^2/2 [tan theta]_0^(arctan R)\ // &=1/2 R^2 tan(arctan R)\ &=1/2 R^3 $